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Factoring Polynomials
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a, b and c are real numbers and a  0.

This is referred to as standard form.

Zero Factor Theorem

If a and b are real numbers and ab = 0, then a = 0 or b = 0.

This theorem is very useful in solving quadratic equations.

Slide 51

Solving Quadratic Equations

Solving Quadratic Equations

Steps for Solving a Quadratic Equation by Factoring

Write the equation in standard form.

Factor the quadratic completely.

Set each factor containing a variable equal to 0.

Solve the resulting equations.

Check each solution in the original equation.

Slide 52

Solve x2  5x = 24.

Solve x2 5x = 24.

First write the quadratic equation in standard form.

x2 5x 24 = 0

Now we factor the quadratic using techniques from the previous sections.

x2 5x 24 = (x 8)(x + 3) = 0

We set each factor equal to 0.

x 8 = 0 or x + 3 = 0, which will simplify to

x = 8 or x = 3

Solving Quadratic Equations

Example

Continued.

Slide 53

Check both possible answers in the original equation.

Check both possible answers in the original equation.

82 5(8) = 64 40 = 24 true

(3)2 5(3) = 9 (15) = 24 true

So our solutions for x are 8 or 3.

Example Continued

Solving Quadratic Equations

Slide 54

Solve 4x(8x + 9) = 5

Solve 4x(8x + 9) = 5

First write the quadratic equation in standard form.

32x2 + 36x = 5

32x2 + 36x 5 = 0

Now we factor the quadratic using techniques from the previous sections.

32x2 + 36x 5 = (8x 1)(4x + 5) = 0

We set each factor equal to 0.

8x 1 = 0 or 4x + 5 = 0

Solving Quadratic Equations

Example

Continued.

Slide 55

Check both possible answers in the original equation.

Check both possible answers in the original equation.

Example Continued

Solving Quadratic Equations

Slide 56

Finding x-intercepts

Finding x-intercepts

Recall that in Chapter 3, we found the x-intercept of linear equations by letting y = 0 and solving for x.

The same method works for x-intercepts in quadratic equations.

Note: When the quadratic equation is written in standard form, the graph is a parabola opening up (when a > 0) or down (when a < 0), where a is the coefficient of the x2 term.

The intercepts will be where the parabola crosses the x-axis.

Slide 57

Find the x-intercepts of the graph of y = 4x2 + 11x + 6.

Find the x-intercepts of the graph of y = 4x2 + 11x + 6.

The equation is already written in standard form, so we let y = 0, then factor the quadratic in x.

0 = 4x2 + 11x + 6 = (4x + 3)(x + 2)

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