Question

polynomials linear factors and zeros practice
problems 1 – 4, write each polynomial in factored form. check by multiplication.

1. $f(x) = x^3 - 5x^2 + 6x$

$x(x^2 - 5x + 6)$
$x(x + 2)(x + 3)$
linear

1. $h(x) = 3x^3 + 5x^2 + 9x + 15$

Explanation:

Problem 2: Factoring \( f(x) = x^3 - 5x^2 + 6x \)

Step 1: Factor out the GCF

The greatest common factor (GCF) of \( x^3 \), \( -5x^2 \), and \( 6x \) is \( x \). Factor out \( x \):
\( f(x) = x(x^2 - 5x + 6) \)

Step 2: Factor the quadratic

Factor \( x^2 - 5x + 6 \). We need two numbers that multiply to \( 6 \) and add to \( -5 \). These numbers are \( -2 \) and \( -3 \):
\( x^2 - 5x + 6 = (x - 2)(x - 3) \)

Step 3: Write the fully factored form

Substitute the factored quadratic back into the expression:
\( f(x) = x(x - 2)(x - 3) \)

(Note: The original handwritten answer had a sign error; it should be \( (x - 2)(x - 3) \) instead of \( (x + 2)(x + 3) \).)

Problem 4: Factoring \( h(x) = 3x^3 + 5x^2 + 9x + 15 \)

Step 1: Group terms

Group the first two terms and the last two terms:
\( h(x) = (3x^3 + 5x^2) + (9x + 15) \)

Step 2: Factor out GCF from each group

Factor \( x^2 \) from the first group and \( 3 \) from the second group:
\( h(x) = x^2(3x + 5) + 3(3x + 5) \)

Step 3: Factor out the common binomial

The common binomial factor is \( (3x + 5) \). Factor it out:
\( h(x) = (x^2 + 3)(3x + 5) \)

Final Answers:

1. \( \boldsymbol{f(x) = x(x - 2)(x - 3)} \)
2. \( \boldsymbol{h(x) = (x^2 + 3)(3x + 5)} \)

Answer:

Step 1: Group terms

Group the first two terms and the last two terms:
\( h(x) = (3x^3 + 5x^2) + (9x + 15) \)

Step 2: Factor out GCF from each group

Factor \( x^2 \) from the first group and \( 3 \) from the second group:
\( h(x) = x^2(3x + 5) + 3(3x + 5) \)

Step 3: Factor out the common binomial

The common binomial factor is \( (3x + 5) \). Factor it out:
\( h(x) = (x^2 + 3)(3x + 5) \)

Final Answers:

1. \( \boldsymbol{f(x) = x(x - 2)(x - 3)} \)
2. \( \boldsymbol{h(x) = (x^2 + 3)(3x + 5)} \)