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All courses Algebra II · F-IF.8 34 of 55
Rewrite functions in equivalent forms to reveal and explain useful properties.

Rewrite a polynomial in factored form to reveal its zeros

Problem
What factored form would reveal the zeros of \(x^2-5x+6\)?
Your answer
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Hint

Factor the quadratic into two binomials. Look for two numbers that multiply to 6 and add to -5.

For x²+bx+c, look for two numbers whose product is c and whose sum is b. Those numbers become the constants in the binomial factors.

Solution walkthrough

01

Set up integer factors

\[x^2-5x+6=(x+m)(x+n),~\text{so}~mn=6~\text{and}~m+n=-5\]

Expanding (x+m)(x+n) gives x²+(m+n)x+mn. Matching coefficients turns the factoring task into a product-and-sum search.

02

Choose the signs and values

\[(-2)(-3)=6~\text{and}~(-2)+(-3)=-5\]

A positive product and negative sum require both integers to be negative. The pair -2 and -3 satisfies both conditions.

03

Factor and verify

\[(x-2)(x-3)=x^2-5x+6\]

Multiplying the factors returns the original trinomial. The factor form reveals zeros x=2 and x=3, so the requested form is (x-2)(x-3).

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Another way

  1. Find the roots with the quadratic formula, x=2 and x=3, then write the corresponding factors x-2 and x-3.

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Common mistake

Do not use (x+2)(x+3). That pair gives the correct constant 6 but the wrong middle coefficient +5.

Solution walkthrough video