maths.free › Algebra › 6. Factoring › Polynomial equations

Polynomial equations

Rational roots, factoring by division and the fundamental theorem of algebra.

A polynomial of degree n has exactly n roots when complex ones and repeats are counted. Find rational roots by testing factors of the constant term over factors of the leading coefficient, divide them out, and finish with a quadratic.

Use the Zero Product Property

We will first solve some quadratic equations by using the Zero Product Property. The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.

We will now use the Zero Product Property, to solve a quadratic equation.

How to Solve a Quadratic Equation Using the Zero Product Property

Try it.

Solve: \((5n-2)(6n-1)=0.\)

Solution

Solve Quadratic Equations by Factoring

The Zero Product Property works very nicely to solve quadratic equations. The quadratic equation must be factored, with zero isolated on one side. So we must be sure to start with the quadratic equation in standard form, \(a{x}^{2}+bx+c=0.\) Then we must factor the expression on the left.

How to Solve a Quadratic Equation by Factoring

Try it.

Solve: \(2{y}^{2}=13y+45.\)

Solution

Before we factor, we must make sure the quadratic equation is in standard form.

Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?

Example

Try it.

Solve: \(169{x}^{2}=49.\)

Solution
\(\ 169{x}^{2}=49\)
Write the quadratic equation in standard form.\(\ 169{x}^{2}-49=0\)
Factor. It is a difference of squares.\(\ (13x-7)(13x+7)=0\)
Use the Zero Product Property to set each factor to \(0\).
Solve each equation.		\(\begin{array}{lllllllll}13x-7 & = & 0 & & & & 13x+7 & = & 0 \\ 13x & = & 7 & & & & 13x & = & -7 \\ x & = & \frac{7}{13} & & & & x & = & -\frac{7}{13}\end{array}\)

Check:

We leave the check up to you.

In the next example, the left side of the equation is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.

Example

Try it.

Solve: \((3x-8)(x-1)=3x.\)

Solution
\(\ (3x-8)(x-1)=3x\)
Multiply the binomials.\(\ 3{x}^{2}-11x+8=3x\)
Write the quadratic equation in standard form.\(\ 3{x}^{2}-14x+8=0\)
Factor the trinomial.\(\ (3x-2)(x-4)=0\)
Use the Zero Product Property to set each factor to 0.
Solve each equation.		\(\begin{array}{lllllllll}3x-2 & = & 0 & & & & x-4 & = & 0 \\ 3x & = & 2 & & & & x & = & 4\end{array}\)
\(\ x=\frac{2}{3}\)
Check your answers.The check is left to you.

In the next example, when we factor the quadratic equation we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Equations with Polynomial Functions

As our study of polynomial functions continues, it will often be important to know when the function will have a certain value or what points lie on the graph of the function. Our work with the Zero Product Property will be help us find these answers.

Example

Try it.

For the function \(f(x)={x}^{2}+2x-2,\)

ⓐ find x when \(f(x)=6\) ⓑ find two points that lie on the graph of the function.

Solution

ⓐ

\(f(x)={x}^{2}+2x-2\)
Substitute 6 for \(f(x)\).\(\ 6={x}^{2}+2x-2\)
Put the quadratic in standard form.\(\ {x}^{2}+2x-8=0\)
Factor the trinomial.\((x+4)(x-2)=0\)
Use the zero product property.
Solve.		\(\begin{array}{lllllllllll}x+4 & = & 0 & & & \text{or} & & & x-2 & = & 0 \\ x & = & -4 & & & \text{or} & & & x & = & 2\end{array}\)
Check:
\(\ \begin{array}{lllll}\ f(x)={x}^{2}+2x-2 & & & & f(x)={x}^{2}+2x-2 \\ f(\text{-}4)={(\text{-}4)}^{2}+2(\text{-}4)-2 & & & & f(2)={2}^{2}+2\cdot 2-2 \\ f(\text{-}4)=16-8-2 & & & & f(2)=4+4-2 \\ f(\text{-}4)=6✓ & & & & f(2)=6✓\end{array}\)


ⓑ Since \(f(-4)=6\) and \(f(2)=6,\) the points \((-4,6)\) and \((2,6)\) lie on the graph of the function.

The Zero Product Property also helps us determine where the function is zero. A value of x where the function is 0, is called a zero of the function.

When \(f(x)=0,\) the point \((x,0)\) is a point on the graph. This point is an x-intercept of the graph. It is often important to know where the graph of a function crosses the axes. We will see some examples later.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Applications Modeled by Polynomial Equations

The problem-solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to polynomial equations. We will copy the problem-solving strategy here so we can use it for reference.

We will start with a number problem to get practice translating words into a polynomial equation.

Example

Try it.

The product of two consecutive odd integers is 323. Find the integers.

Solution
Step 1. Read the problem.
Step 2. Identify what we are looking for.We are looking for two consecutive integers.
Step 3. Name what we are looking for.Let \(n=\ \text{the first integer.}\)
\(n+2=\ \text{next consecutive odd integer}\)
Step 4. Translate into an equation. Restate the problem in a sentence.The product of the two consecutive odd integers is 323.
\(\ n(n+2)=323\)
Step 5. Solve the equation.\(\ {n}^{2}+2n=323\)
Bring all the terms to one side.\(\ {n}^{2}+2n-323=0\)
Factor the trinomial.\((n-17)(n+19)=0\)
Use the Zero Product Property.
Solve the equations.		\(\begin{array}{lllllllll}n-17 & = & 0 & & & & n+19 & = & 0 \\ n & = & 17 & & & & n & = & -19\end{array}\)
There are two values for n that are solutions to this problem. So there are two sets of consecutive odd integers that will work.
If the first integer is \(n=17\)If the first integer is \(n=-19\)
then the next odd integer isthen the next odd integer is
\(n+2\)\(n+2\)
\(17+2\)\(-19+2\)
\(19\)\(-17\)
\(17,19\)\(-17,-19\)
Step 6. Check the answer.
The results are consecutive odd integers
\(17,19\ \text{and}\ \text{-}19,-17.\)
\(17\cdot 19=323✓\ \text{-}19(\text{-}17)=323✓\)
Both pairs of consecutive integers are solutions.
Step 7. Answer the questionThe consecutive integers are 17, 19 and \(-19,-17.\)

In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.

We will use this formula in the next example.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

  • Polynomial Equation: A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
  • Quadratic Equation: An equation of the form \(a{x}^{2}+bx+c=0\) is called a quadratic equation.
    \[a,b,c\ \text{are real numbers and}\ a\ne 0\]
  • Zero Product Property: If \(a\cdot b=0,\) then either \(a=0\) or \(b=0\) or both.
  • How to use the Zero Product Property
    1. Set each factor equal to zero.
    2. Solve the linear equations.
    3. Check.
  • How to solve a quadratic equation by factoring.
    1. Write the quadratic equation in standard form, \(a{x}^{2}+bx+c=0.\)
    2. Factor the quadratic expression.
    3. Use the Zero Product Property.
    4. Solve the linear equations.
    5. Check. Substitute each solution separately into the original equation.
  • Zero of a Function: For any function f, if \(f(x)=0,\) then x is a zero of the function.
  • How to use a problem solving strategy to solve word problems.
    1. Read the problem. Make sure all the words and ideas are understood.
    2. Identify what we are looking for.
    3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
    5. Solve the equation using appropriate algebra techniques.
    6. Check the answer in the problem and make sure it makes sense.
    7. Answer the question with a complete sentence.

Radni primjer: x^3 - 6x^2 + 11x - 6 = 0

Algebra · Polynomial-Equations

Solve x^3 - 6x^2 + 11x - 6 = 0

x^{3} - 6 x^{2} + 11 x - 6 = 0

Korak po korak

  1. x^{3} - 6 x^{2} + 11 x - 6 = 0

    Start from the equation as given.

  2. \left(x - 3\right) \left(x - 2\right) \left(x - 1\right) = 0

    Factor the polynomial (rational-root test + division).

  3. x - 1 = 0 \;\Rightarrow\; 1

    Set each factor equal to zero and solve it.

  4. x - 3 = 0 \;\Rightarrow\; 3

    Set each factor equal to zero and solve it.

  5. x - 2 = 0 \;\Rightarrow\; 2

    Set each factor equal to zero and solve it.

  6. x = 1 ,\; x = 3 ,\; x = 2

    All 3 roots.

Otkrij odgovor
x = 1 ,\; x = 3 ,\; x = 2

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Solve: \(5y-3=0.\)
    If you missed this problem, review .

    Otkrij odgovor

    \(y=\frac{3}{5}\)

  2. Factor completely: \({n}^{3}-9{n}^{2}-22n.\)
    If you missed this problem, review .

    Otkrij odgovor

    \(n(n-11)(n+2)\)

  3. If \(f(x)=8x-16,\) find \(f(3)\) and solve \(f(x)=0.\)
    If you missed this problem, review .

    Otkrij odgovor

    \(8;\ x=2\)

  4. Solve: \((5n-2)(6n-1)=0.\)

  5. Solve: \((3m-2)(2m+1)=0.\)

    Otkrij odgovor

    \(m=\frac{2}{3},m=-\frac{1}{2}\)

  6. Solve: \((4p+3)(4p-3)=0.\)

    Otkrij odgovor

    \(p=-\frac{3}{4},p=\frac{3}{4}\)

  7. Solve: \(2{y}^{2}=13y+45.\)

  8. Solve: \(3{c}^{2}=10c-8.\)

    Otkrij odgovor

    \(c=2,c=\frac{4}{3}\)

  9. Solve: \(2{d}^{2}-5d=3.\)

    Otkrij odgovor

    \(d=3,d=-\frac{1}{2}\)

  10. Solve: \(169{x}^{2}=49.\)

    Otkrij odgovor
    \(\ 169{x}^{2}=49\)
    Write the quadratic equation in standard form.\(\ 169{x}^{2}-49=0\)
    Factor. It is a difference of squares.\(\ (13x-7)(13x+7)=0\)
    Use the Zero Product Property to set each factor to \(0\).
    Solve each equation.    		\(\begin{array}{lllllllll}13x-7 & = & 0 & & & & 13x+7 & = & 0 \\ 13x & = & 7 & & & & 13x & = & -7 \\ x & = & \frac{7}{13} & & & & x & = & -\frac{7}{13}\end{array}\)

    Check:

    We leave the check up to you.

  11. Solve: \(25{p}^{2}=49.\)

    Otkrij odgovor

    \(p=\frac{7}{5},\text{p}=-\frac{7}{5}\)

  12. Solve: \(36{x}^{2}=121.\)

    Otkrij odgovor

    \(x=\frac{11}{6},x=-\frac{11}{6}\)

  13. Solve: \((3x-8)(x-1)=3x.\)

    Otkrij odgovor
    \(\ (3x-8)(x-1)=3x\)
    Multiply the binomials.\(\ 3{x}^{2}-11x+8=3x\)
    Write the quadratic equation in standard form.\(\ 3{x}^{2}-14x+8=0\)
    Factor the trinomial.\(\ (3x-2)(x-4)=0\)
    Use the Zero Product Property to set each factor to 0.
    Solve each equation.    		\(\begin{array}{lllllllll}3x-2 & = & 0 & & & & x-4 & = & 0 \\ 3x & = & 2 & & & & x & = & 4\end{array}\)
    \(\ x=\frac{2}{3}\)
    Check your answers.The check is left to you.

  14. Solve: \((2m+1)(m+3)=12m.\)

    Otkrij odgovor

    \(m=1,m=\frac{3}{2}\)

  15. Solve: \((k+1)(k-1)=8.\)

    Otkrij odgovor

    \(k=3,k=-3\)

  16. Solve: \(3{x}^{2}=12x+63.\)

    Otkrij odgovor
    \(\ 3{x}^{2}=12x+63\)
    Write the quadratic equation in standard form.\(\ 3{x}^{2}-12x-63=0\)
    Factor the greatest common factor first.\(\ 3({x}^{2}-4x-21)=0\)
    Factor the trinomial.\(\ 3(x-7)(x+3)=0\)
    Use the Zero Product Property to set each factor to 0.
    Solve each equation.    		\(\ \begin{array}{lllllllllllll}3\ne 0 & & & & x-7 & = & 0 & & & & x+3 & = & 0 \\ 3\ne 0 & & & & x & = & 7 & & & & x & = & -3\end{array}\)
    Check your answers.The check is left to you.

  17. Solve: \(18{a}^{2}-30=-33a.\)

    Otkrij odgovor

    \(a=-\frac{5}{2},a=\frac{2}{3}\)

  18. Solve: \(123b=-6-60{b}^{2}.\)

    Otkrij odgovor

    \(b=-2,b=-\frac{1}{20}\)

  19. Solve: \(9{m}^{3}+100m=60{m}^{2}.\)

    Otkrij odgovor
    \(\ 9{m}^{3}+100m=60{m}^{2}\)
    Bring all the terms to one side so that the other side is zero.\(\ 9{m}^{3}-60{m}^{2}+100m=0\)
    Factor the greatest common factor first.\(\ m(9{m}^{2}-60m+100)=0\)
    Factor the trinomial.\(\ m(3m-10)(3m-10)=0\)
    Use the Zero Product Property to set each factor to 0.
    Solve each equation.    		\(\ \begin{array}{lllllllllllll}m=0 & & & & 3m-10 & = & 0 & & & & 3m-10 & = & 0 \\ m=0 & & & & m & = & \frac{10}{3} & & & & m & = & \frac{10}{3}\end{array}\)
    Check your answers.The check is left to you.

  20. Solve: \(8{x}^{3}=24{x}^{2}-18x.\)

    Otkrij odgovor

    \(x=0,x=\frac{3}{2}\)

  21. Solve: \(16{y}^{2}=32{y}^{3}+2y.\)

    Otkrij odgovor

    \(y=0,y=\frac{1}{4}\)

  22. For the function \(f(x)={x}^{2}+2x-2,\)

    ⓐ find x when \(f(x)=6\) ⓑ find two points that lie on the graph of the function.

    Otkrij odgovor

    ⓐ

    \(f(x)={x}^{2}+2x-2\)
    Substitute 6 for \(f(x)\).\(\ 6={x}^{2}+2x-2\)
    Put the quadratic in standard form.\(\ {x}^{2}+2x-8=0\)
    Factor the trinomial.\((x+4)(x-2)=0\)
    Use the zero product property.
    Solve.    		\(\begin{array}{lllllllllll}x+4 & = & 0 & & & \text{or} & & & x-2 & = & 0 \\ x & = & -4 & & & \text{or} & & & x & = & 2\end{array}\)
    Check:
    \(\ \begin{array}{lllll}\ f(x)={x}^{2}+2x-2 & & & & f(x)={x}^{2}+2x-2 \\ f(\text{-}4)={(\text{-}4)}^{2}+2(\text{-}4)-2 & & & & f(2)={2}^{2}+2\cdot 2-2 \\ f(\text{-}4)=16-8-2 & & & & f(2)=4+4-2 \\ f(\text{-}4)=6✓ & & & & f(2)=6✓\end{array}\)


    ⓑ Since \(f(-4)=6\) and \(f(2)=6,\) the points \((-4,6)\) and \((2,6)\) lie on the graph of the function.

  23. For the function \(f(x)={x}^{2}-2x-8,\)

    ⓐ find x when \(f(x)=7\) ⓑ Find two points that lie on the graph of the function.

    Otkrij odgovor

    ⓐ \(x=-3\) or \(x=5\)
    ⓑ \((-3,7)\) \((5,7)\)

  24. For the function \(f(x)={x}^{2}-8x+3,\)

    ⓐ find x when \(f(x)=-4\) ⓑ Find two points that lie on the graph of the function.

    Otkrij odgovor

    ⓐ \(x=1\) or \(x=7\)
    ⓑ \((1,-4)\) \((7,-4)\)

  25. For the function \(f(x)=3{x}^{2}+10x-8,\) find

    ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function

    Otkrij odgovor

    ⓐ To find the zeros of the function, we need to find when the function value is 0.

    \(f(x)=3{x}^{2}+10x-8\)
    Substitute 0 for \(f(x)\).\(\ 0=3{x}^{2}+10x-8\)
    Factor the trinomial.\((x+4)(3x-2)=0\)
    Use the zero product property.
    Solve.    		\(\begin{array}{lllllllllll}x+4 & = & 0 & & & \text{or} & & & 3x-2 & = & 0 \\ x & = & -4 & & & \text{or} & & & x & = & \frac{2}{3}\end{array}\)


    ⓑ An x-intercept occurs when \(y=0.\) Since \(f(-4)=0\) and \(f(\frac{2}{3})=0,\) the points \((-4,0)\) and \((\frac{2}{3},0)\) lie on the graph. These points are x-intercepts of the function.


    ⓒ A y-intercept occurs when \(x=0.\) To find the y-intercepts we need to find \(f(0).\)
    \(f(x)=3{x}^{2}+10x-8\)
    Find \(f(0)\) by substituting 0 for \(x\).\(f(0)=3\cdot {0}^{2}+10\cdot 0-8\)
    Simplify.\(f(0)=-8\)

    Since \(f(0)=-8,\) the point \((0,-8)\) lies on the graph. This point is the y-intercept of the function.

  26. For the function \(f(x)=2{x}^{2}-7x+5,\) find

    ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function.

    Otkrij odgovor

    ⓐ \(x=1\) or \(x=\frac{5}{2}\)
    ⓑ \((1,0),\) \((\frac{5}{2},0)\) ⓒ \((0,5)\)

  27. For the function \(f(x)=6{x}^{2}+13x-15,\) find

    ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function.

    Otkrij odgovor

    ⓐ \(x=-3\) or \(x=\frac{5}{6}\)
    ⓑ \((-3,0),\) \((\frac{5}{6},0)\) ⓒ \((0,-15)\)

  28. The product of two consecutive odd integers is 323. Find the integers.

    Otkrij odgovor
    Step 1. Read the problem.
    Step 2. Identify what we are looking for.We are looking for two consecutive integers.
    Step 3. Name what we are looking for.Let \(n=\ \text{the first integer.}\)
    \(n+2=\ \text{next consecutive odd integer}\)
    Step 4. Translate into an equation. Restate the problem in a sentence.The product of the two consecutive odd integers is 323.
    \(\ n(n+2)=323\)
    Step 5. Solve the equation.\(\ {n}^{2}+2n=323\)
    Bring all the terms to one side.\(\ {n}^{2}+2n-323=0\)
    Factor the trinomial.\((n-17)(n+19)=0\)
    Use the Zero Product Property.
    Solve the equations.    		\(\begin{array}{lllllllll}n-17 & = & 0 & & & & n+19 & = & 0 \\ n & = & 17 & & & & n & = & -19\end{array}\)
    There are two values for n that are solutions to this problem. So there are two sets of consecutive odd integers that will work.
    If the first integer is \(n=17\)If the first integer is \(n=-19\)
    then the next odd integer isthen the next odd integer is
    \(n+2\)\(n+2\)
    \(17+2\)\(-19+2\)
    \(19\)\(-17\)
    \(17,19\)\(-17,-19\)
    Step 6. Check the answer.
    The results are consecutive odd integers
    \(17,19\ \text{and}\ \text{-}19,-17.\)
    \(17\cdot 19=323✓\ \text{-}19(\text{-}17)=323✓\)
    Both pairs of consecutive integers are solutions.
    Step 7. Answer the questionThe consecutive integers are 17, 19 and \(-19,-17.\)

  29. The product of two consecutive odd integers is 255. Find the integers.

    Otkrij odgovor

    \(-15,-17\) and 15, 17

  30. The product of two consecutive odd integers is 483 Find the integers.

    Otkrij odgovor

    \(-23,-21\) and 21, 23

  31. A rectangular bedroom has an area 117 square feet. The length of the bedroom is four feet more than the width. Find the length and width of the bedroom.

    Otkrij odgovor

    Step 1. Read the problem. In problems involving
    geometric figures, a sketch can help you visualize
    the situation.
    Step 2. Identify what you are looking for.We are looking for the length and width.
    Step 3. Name what you are looking for.Let \(w=\) the width of the bedroom.
    The length is four feet more than the width.\(w+4=\) the length of the garden
    Step 4. Translate into an equation.
    Restate the important information in a sentence.The area of the bedroom is 117 square feet.
    Use the formula for the area of a rectangle.\(A=l\cdot w\\)
    Substitute in the variables.\(\ 117=(w+4)w\)
    Step 5. Solve the equation Distribute first.\(117={w}^{2}+4w\)
    Get zero on one side.\(117={w}^{2}+4w\)
    Factor the trinomial.\(\ 0={w}^{2}+4w-117\)
    Use the Zero Product Property.\(\ 0=(w+13)(w-9)\)
    Solve each equation.\(\ 0=w+13\ 0=w-9\)
    Since w is the width of the bedroom, it does not
    make sense for it to be negative. We eliminate that value for w.    		\(\ -13=w\ 9=w\)
    \(w=9\) Width is 9 feet.     
    Find the value of the length.\(w+4\\)
    \(9+4\\)
    \(\\)13  Length is 13 feet.
    Step 6. Check the answer.
    Does the answer make sense?


    Yes, this makes sense.
    Step 7. Answer the question.The width of the bedroom is 9 feet and
    the length is 13 feet.

  32. A rectangular sign has an area of 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.

    Otkrij odgovor

    The width is 5 feet and length is 6 feet.

  33. A rectangular patio has an area of 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.

    Otkrij odgovor

    The width of the patio is 12 feet and the length is 15 feet.

  34. A boat’s sail is in the shape of a right triangle as shown. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the other side. Find the lengths of the sides of the sail.

    Otkrij odgovor

    Step 1. Read the problem
    Step 2. Identify what you are looking for.We are looking for the lengths of the
    sides of the sail.
    Step 3. Name what you are looking for.
    One side is 7 less than the other.    		Let \(x=\) length of a side of the sail.
    \(x-7=\) length of other side
    Step 4. Translate into an equation. Since this is a
    right triangle we can use the Pythagorean Theorem.    		\(\ {a}^{2}+{b}^{2}={c}^{2}\)
    Substitute in the variables.\(\ {x}^{2}+{(x-7)}^{2}={17}^{2}\)
    Step 5. Solve the equation
    Simplify.    		\({x}^{2}+{x}^{2}-14x+49=289\)
    \(\ 2{x}^{2}-14x+49=289\)
    It is a quadratic equation, so get zero on one side.\(\ 2{x}^{2}-14x-240=0\)
    Factor the greatest common factor.\(\ 2({x}^{2}-7x-120)=0\)
    Factor the trinomial.\(\ 2(x-15)(x+8)=0\)
    Use the Zero Product Property.\(2\ne 0\ x-15=0\ x+8=0\)
    Solve.\(2\ne 0\ x=15\ x=-8\)
    Since x is a side of the triangle, \(x=-8\) does not
    make sense.    		\(2\ne 0\ x=15\ x=-8\)
    Find the length of the other side.
       If the length of one side is
       then the length of the other side is


    8 is the length of the other side.
    Step 6. Check the answer in the problem
    Do these numbers make sense?

    Step 7. Answer the questionThe sides of the sail are 8, 15 and 17 feet.

  35. Justine wants to put a deck in the corner of her backyard in the shape of a right triangle. The length of one side of the deck is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the deck.

    Otkrij odgovor

    5 feet and 12 feet

  36. A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of the other leg. Find the lengths of the hypotenuse and the other leg.

    Otkrij odgovor

    The other leg is 24 feet and the hypotenuse is 25 feet.

  37. Dennis is going to throw his rubber band ball upward from the top of a campus building. When he throws the rubber band ball from 80 feet above the ground, the function \(h(t)=-16{t}^{2}+64t+80\) models the height, h, of the ball above the ground as a function of time, t. Find:

    ⓐ the zeros of this function which tell us when the ball hits the ground, ⓑ when the ball will be 80 feet above the ground, ⓒ the height of the ball at \(t=2\) seconds.

    Otkrij odgovor

    ⓐ The zeros of this function are found by solving \(h(t)=0.\) This will tell us when the ball will hit the ground.

    \(h(t)=0\\)
    Substitute in the polynomial for \(h(t).\)\(-16{t}^{2}+64t+80=0\\)
    Factor the GCF, \(-16.\)\(-16({t}^{2}-4t-5)=0\\)
    Factor the trinomial.\(-16(t-5)(t+1)=0\\)
    Use the Zero Product Property.
    Solve.    		\(\begin{array}{lllllllllll}t-5 & = & 0 & & & & & & \ t+1 & = & 0 \\ t & = & 5 & & & & & & \ t & = & -1\end{array}\)

    The result \(t=5\) tells us the ball will hit the ground 5 seconds after it is thrown. Since time cannot be negative, the result \(t=-1\) is discarded.

    ⓑ The ball will be 80 feet above the ground when \(h(t)=80.\)

    \(h(t)=80\\)
    Substitute in the polynomial for \(h(t).\)\(-16{t}^{2}+64t+80=80\\)
    Subtract 80 from both sides.\(-16{t}^{2}+64t=0\\)
    Factor the GCF, \(-16t.\)\(-16t(t-4)=0\\)
    Use the Zero Product Property.
    Solve.    		\(\begin{array}{lllllllll}-16t & = & 0 & & & & t-4 & = & 0 \\ t & = & 0 & & & & t & = & 4\end{array}\\)
    The ball will be at 80 feet the moment Dennis tosses the ball and then 4 seconds later, when the ball is falling.

    ⓒ To find the height ball at \(t=2\) seconds we find \(h(2).\)

    \(h(t)=-16{t}^{2}+64t+80\)
    To find \(h(2)\) substitute 2 for \(t.\)\(h(2)=-16{(2)}^{2}+64\cdot 2+80\)
    Simplify.\(h(2)=144\)
    After 2 seconds, the ball will be at 144 feet.

  38. Genevieve is going to throw a rock from the top a trail overlooking the ocean. When she throws the rock upward from 160 feet above the ocean, the function \(h(t)=-16{t}^{2}+48t+160\) models the height, h, of the rock above the ocean as a function of time, t. Find:

    ⓐ the zeros of this function which tell us when the rock will hit the ocean, ⓑ when the rock will be 160 feet above the ocean, ⓒ the height of the rock at \(t=1.5\) seconds.

    Otkrij odgovor

    ⓐ 5 seconds; ⓑ 0 and 3 seconds; ⓒ 196 feet

  39. Calib is going to throw his lucky penny from his balcony on a cruise ship. When he throws the penny upward from 128 feet above the ground, the function \(h(t)=-16{t}^{2}+32t+128\) models the height, h, of the penny above the ocean as a function of time, t. Find:

    ⓐ the zeros of this function which is when the penny will hit the ocean, ⓑ when the penny will be 128 feet above the ocean, ⓒ the height the penny will be at \(t=1\) seconds which is when the penny will be at its highest point.

    Otkrij odgovor

    ⓐ 4 seconds; ⓑ 0 and 2 seconds; ⓒ 144 feet

  40. \((3a-10)(2a-7)=0\)

    Otkrij odgovor

    \(a=10\text{/}3,a=7\text{/}2\)

Symbols used here

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\neq
not equal
The two sides are different.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Polynomial equations

  1. Use the Zero Product Property
  2. Solve quadratic equations by factoring
  3. Solve equations with polynomial functions
  4. Solve applications modeled by polynomial equations
  5. Set each factor equal to zero.
  6. Solve the linear equations.
  7. Check.
  8. Write the quadratic equation in standard form,

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Pokušaj i ti

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Više u Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsAbsolute value