Answer :
To factor the polynomial [tex]\( f(x) = 4x^3 + 25x^2 - 93x + 54 \)[/tex] into linear factors, knowing that -9 is a zero, we can proceed through the following steps:
1. Use the Zero: Since -9 is a zero of the polynomial, [tex]\( (x + 9) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
2. Divide the Polynomial:
- Perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( (x + 9) \)[/tex] to find the other factors.
- When dividing, you'll find that:
[tex]\[
\frac{4x^3 + 25x^2 - 93x + 54}{x + 9} = 4x^2 - 11x + 6
\][/tex]
This means [tex]\( f(x) = (x + 9)(4x^2 - 11x + 6) \)[/tex].
3. Factor the Quadratic:
- Now, factor the quadratic polynomial [tex]\( 4x^2 - 11x + 6 \)[/tex].
- Look for two numbers that multiply to [tex]\( 4 \times 6 = 24 \)[/tex] and add up to [tex]\(-11\)[/tex]. These numbers are [tex]\(-8\)[/tex] and [tex]\(-3\)[/tex].
- Rewrite [tex]\( -11x \)[/tex] using these numbers: [tex]\( 4x^2 - 8x - 3x + 6 \)[/tex].
- Group and factor by grouping:
[tex]\[
4x^2 - 8x - 3x + 6 = (4x^2 - 8x) + (-3x + 6)
\][/tex]
[tex]\[
= 4x(x - 2) - 3(x - 2)
\][/tex]
[tex]\[
= (4x - 3)(x - 2)
\][/tex]
4. Combine All Factors:
- So, the complete factorization of [tex]\( f(x) \)[/tex] into linear factors is:
[tex]\[
f(x) = (x + 9)(4x - 3)(x - 2)
\][/tex]
These are the linear factors of the given polynomial: [tex]\( (x - 2) \)[/tex], [tex]\( (x + 9) \)[/tex], and [tex]\( (4x - 3) \)[/tex].
1. Use the Zero: Since -9 is a zero of the polynomial, [tex]\( (x + 9) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
2. Divide the Polynomial:
- Perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( (x + 9) \)[/tex] to find the other factors.
- When dividing, you'll find that:
[tex]\[
\frac{4x^3 + 25x^2 - 93x + 54}{x + 9} = 4x^2 - 11x + 6
\][/tex]
This means [tex]\( f(x) = (x + 9)(4x^2 - 11x + 6) \)[/tex].
3. Factor the Quadratic:
- Now, factor the quadratic polynomial [tex]\( 4x^2 - 11x + 6 \)[/tex].
- Look for two numbers that multiply to [tex]\( 4 \times 6 = 24 \)[/tex] and add up to [tex]\(-11\)[/tex]. These numbers are [tex]\(-8\)[/tex] and [tex]\(-3\)[/tex].
- Rewrite [tex]\( -11x \)[/tex] using these numbers: [tex]\( 4x^2 - 8x - 3x + 6 \)[/tex].
- Group and factor by grouping:
[tex]\[
4x^2 - 8x - 3x + 6 = (4x^2 - 8x) + (-3x + 6)
\][/tex]
[tex]\[
= 4x(x - 2) - 3(x - 2)
\][/tex]
[tex]\[
= (4x - 3)(x - 2)
\][/tex]
4. Combine All Factors:
- So, the complete factorization of [tex]\( f(x) \)[/tex] into linear factors is:
[tex]\[
f(x) = (x + 9)(4x - 3)(x - 2)
\][/tex]
These are the linear factors of the given polynomial: [tex]\( (x - 2) \)[/tex], [tex]\( (x + 9) \)[/tex], and [tex]\( (4x - 3) \)[/tex].
