Question

1. divide the following polynomial functions by the linear factor provided using an area model.

a. $f(x) = x^3 + 9x^2 - 10x - 168$, $(x - 4)$
b. $g(x) = 5x^3 - 15x^2 + 12x - 20$, $(x + 6)$
c. $q(x) = 6x^3 - 2x + 3x^4 + 4 - 5x^2$, $(x + 2)$
d. $j(x) = 5x - 2x^2 - 8 + x^3$, $(x - 3)$

Explanation:

Step1: Arrange Polynomial Terms

For \( f(x) = x^3 + 9x^2 - 10x - 168 \), the terms are already in descending order of exponents. The divisor is \( (x - 4) \). We set up the area model with one side as \( (x - 4) \) and we need to find the other side (the quotient) such that the product of the two sides is \( f(x) \).

Step2: Find the First Term of the Quotient

Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \), we get \( x^2 \). Multiply \( (x - 4) \) by \( x^2 \), we get \( x^3 - 4x^2 \). Subtract this from the dividend:
\( (x^3 + 9x^2 - 10x - 168) - (x^3 - 4x^2) = 13x^2 - 10x - 168 \)

Step3: Find the Second Term of the Quotient

Divide the leading term of the new dividend \( 13x^2 \) by \( x \), we get \( 13x \). Multiply \( (x - 4) \) by \( 13x \), we get \( 13x^2 - 52x \). Subtract this from the new dividend:
\( (13x^2 - 10x - 168) - (13x^2 - 52x) = 42x - 168 \)

Step4: Find the Third Term of the Quotient

Divide the leading term of the new dividend \( 42x \) by \( x \), we get \( 42 \). Multiply \( (x - 4) \) by \( 42 \), we get \( 42x - 168 \). Subtract this from the new dividend:
\( (42x - 168) - (42x - 168) = 0 \)
Since there is no remainder, the quotient is \( x^2 + 13x + 42 \).

(We can follow similar steps for other sub - questions. For example, for sub - question d:

Step1: Arrange Polynomial Terms

For \( j(x)=x^{3}-2x^{2}+5x - 8 \) (arranged in descending order of exponents) and divisor \( (x - 3) \)

Step2: Find the First Term of the Quotient

Divide \( x^{3}\) by \( x\) to get \( x^{2}\). Multiply \( (x - 3)\) by \( x^{2}\) to get \( x^{3}-3x^{2}\). Subtract from the dividend:
\((x^{3}-2x^{2}+5x - 8)-(x^{3}-3x^{2})=x^{2}+5x - 8\)

Step3: Find the Second Term of the Quotient

Divide \( x^{2}\) by \( x\) to get \( x\). Multiply \( (x - 3)\) by \( x\) to get \( x^{2}-3x\). Subtract from the new dividend:
\((x^{2}+5x - 8)-(x^{2}-3x)=8x - 8\)

Step4: Find the Third Term of the Quotient

Divide \( 8x\) by \( x\) to get \( 8\). Multiply \( (x - 3)\) by \( 8\) to get \( 8x - 24\). Subtract from the new dividend:
\((8x - 8)-(8x - 24)=16\)
So the quotient is \( x^{2}+x + 8\) with a remainder of \( 16\))

Answer:

For part a: The quotient is \( x^{2}+13x + 42\) (with remainder \( 0\))
For part d: The quotient is \( x^{2}+x + 8\) with remainder \( 16\)

(Note: For parts b and c, we can follow the same area - model polynomial division steps. For part b, when we perform the division of \( 5x^{3}-15x^{2}+12x - 20\) by \( (x + 6)\), we will find that there will be a non - zero remainder. For part c, first, we need to arrange the polynomial \( q(x)=3x^{4}+6x^{3}-5x^{2}-2x + 4\) in descending order and then perform the division by \( (x + 2)\) using the area model.)