Question

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fill in the missing values below one at a time to find the quotient when $3x^3 - 19x^2 + 24x + 16$ is divided by $x - 4$.
chart with rows labeled $x$ and $-4$, columns with empty boxes and one box with $3x^3$
answer
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Explanation:

Step1: Identify the quotient form

We are dividing a cubic polynomial \(3x^3 - 19x^2 + 24x + 16\) by a linear polynomial \(x - 4\). The quotient should be a quadratic polynomial, let's assume the quotient is \(ax^2+bx + c\). So, we can express the division as \((x - 4)(ax^2+bx + c)=3x^3 - 19x^2 + 24x + 16\).

Step2: Find the coefficient \(a\)

Multiply \(x\) with \(ax^2\) to get \(ax^3\). Comparing with the leading term \(3x^3\) of the dividend, we have \(a = 3\). So the first term of the quotient is \(3x^2\).

Step3: Find the coefficient \(b\)

Now, multiply \(x\) with \(bx\) to get \(bx^2\) and multiply \(- 4\) with \(ax^2=3x^2\) to get \(- 4\times3x^2=-12x^2\). The sum of the \(x^2\) terms in the product \((x - 4)(3x^2+bx + c)\) is \(bx^2-12x^2=(b - 12)x^2\). Comparing with the \(x^2\) term \(-19x^2\) of the dividend, we have \(b-12=-19\), so \(b=-19 + 12=-7\). So the second term of the quotient is \(-7x\).

Step4: Find the coefficient \(c\)

Multiply \(x\) with \(c\) to get \(cx\) and multiply \(-4\) with \(bx=-7x\) to get \(-4\times(-7x) = 28x\). The sum of the \(x\) terms in the product \((x - 4)(3x^2-7x + c)\) is \(cx + 28x=(c + 28)x\). Comparing with the \(x\) term \(24x\) of the dividend, we have \(c + 28=24\), so \(c=24 - 28=-4\). Wait, but let's check the constant term. Multiply \(-4\) with \(c\) to get \(-4c\). The constant term of the dividend is \(16\), so \(-4c = 16\), which gives \(c=-4\). Let's verify the product:

\((x - 4)(3x^2-7x-4)=x(3x^2-7x - 4)-4(3x^2-7x - 4)\)

\(=3x^3-7x^2-4x-12x^2 + 28x + 16\)

\(=3x^3-(7x^2 + 12x^2)+(28x-4x)+16\)

\(=3x^3-19x^2 + 24x + 16\), which matches the dividend.

Now, let's fill the table:

• The top row (quotient terms): \(3x^2\), \(-7x\), \(-4\)
• The middle row (product of \(x\) and quotient terms):
• \(x\times3x^2 = 3x^3\) (already given)
• \(x\times(-7x)=-7x^2\)
• \(x\times(-4)=-4x\)
• The bottom row (product of \(-4\) and quotient terms):
• \(-4\times3x^2=-12x^2\)
• \(-4\times(-7x)=28x\)
• \(-4\times(-4) = 16\)

Answer:

Top row (quotient terms): \(3x^2\), \(-7x\), \(-4\)

Middle row (product of \(x\) and quotient terms): \(3x^3\) (given), \(-7x^2\), \(-4x\)

Bottom row (product of \(-4\) and quotient terms): \(-12x^2\), \(28x\), \(16\)

The quotient is \(3x^2-7x - 4\)