Solve Equations by Factoring

Section 6.4

MATH 116-460

Mr. Keltner

Solving Polynomial Equations by Factoring

In this chapter, we have worked with strategies for factoring polynomial expressions.

In this section, we extend those strategies to include solving equations involving polynomials.

The principle of solving polynomial equations by factoring involves something called the Zero-Factor Theorem.

Zero-Factor Theorem The Zero-Factor Theorem states that if

the product of two or more factors is 0, then at least one of the factors is equal to 0.

In symbols, it looks like this:– If a•b = 0, then either a = 0 or b = 0 or both

a and b equal 0.

Example 1 Solve the equation (n - 6)(n + 1) = 0.

Polynomial Equations To solve a polynomial equation, arrange

the equation in standard form, which means one side is equal to 0.– Once it is set equal to 0, write in factored

form so we can use the zero-factor theorem.

Checking your answers will verify you have factored correctly.

Example 2Solve 12y2 - 2 = 5y.

Bigger than an x2 term The zero-factor theorem can be used to

solve polynomial equations of degree greater than 2.

It is still a good idea to look for a monomial that can be factored out of all terms in the polynomial.

Example 3: Solve 12x3 + 20x2 - 8x = 0.

Example 4 Solve each equation.

(y - 4)2 = 1

n(n2 - 13) = 3n

Make sure to set equal to zero first.

1/4(x2+8) + 2x=1/8(x-2)

h2(2h + 1) - 3=18h + 6

Applications for Polynomial Equations Some applications for polynomial

equations can also use the zero-factor theorem.

Setting up an equation can also involve finding factors that can be solved for a particular variable.

Try the next example as an application for a polynomial equation.

Example 5:Goin’ to the park

Pilla Park

City Pool Park40 feet

60 feet

x feet

x feet

Suppose the playground by the Eudora City Pool is 60 feet wide and 40 feet long. The area of Pilla Park, the playground by Eudora City Hall, is twice that size (area), but has the same shape. Find the dimensions of that playground.

Using intercepts to graph polynomial functions Remember that:

– A y-intercept occurs where x = 0– A x-intercept occurs where y = 0

Use these tips to find the x- and y-intercepts for polynomial functions and graph them.

As always, the more points you calculate and plot, the more accurate your graph becomes.

Extension of Section 5.2:Graphs of Polynomials Quadratic Functions, in the

formf(x) = ax2 + bx + c,

will have at most two x-intercepts.

Cubic Functions, in the formf(x) = ax3 + bx2 + cx + d,

will have at most three x-intercepts.

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NOx-intercepts

ONEx-intercept

ONEx-intercept

TWOx-intercepts

TWOx-intercepts

THREEx-intercepts

From Factored Form to Finding Intercepts

When a function is in factored form, we can easily find its x-intercepts by applying the Zero-Factor Theorem.

Example 6: Find the x-intercepts of these functions by writing them as a product of factors. f(x) = (x + 2)2(x - 3) f(x) = x2 +2x - 3

Assessment

Pgs. 427-428:

#’s 7-56, multiples of 7