Expert Answer to "a) Using the remainder theorem, determine whether (x−4)..."

High School
Algebra
Pre-Algebra
Factors and multiples

Sinead Mcgee

Answered question

2021-07-30

a) Using the remainder theorem, determine whether

(x - 4)

and

(x - 1)

are factors of the expression

x^{3} + 3 x^{2} - 22 x - 24

.
b) Hence, by use of long division, find all remaining factors of the expression.

Answer & Explanation

Dora

Skilled2021-07-31Added 98 answers

Step 1
Given: $x^{3} + 3 x^{2} - 22 x - 24$
a) $(x - 4)$ Factor $x - 4 = 0$
$x - 4$
${(4)}^{3} + 3 {(4)}^{2} - 22 (4) - 24$
$= 64 + 48 - 88 - 24$
$= 112 - 112 = 0$
$(x - 1)$ Factor $x - 1 = 0$
$x = 1$
${(1)}^{3} + 3 {(1)}^{2} - 22 (1) - 24$
$= 1 + 3 - 22 - 24$
$= 4 - 46$
$= - 40$ (not a factor)
Step 2
b) $(x - 4)$ Factor of $x^{3} + 3 x^{2} - 22 n - 24$
$x = 4$
$\begin{array}{cc} 4 & 1 & 3 & - 22 & - 24 \\ 0 & 4 & 28 & 24 \\ 1 & 7 & 6 & 0 \end{array}$
$x^{2} + 7 x + 6$
$= x^{2} + 6 x + x + 6$
$= x (x + 6) + 1 (x + 6)$
$= (x + 6) (x + 1)$
$(x + 6) (x + 1)$ are the remaining factors.

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