Question

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1 fill in the blank 3 points
the expression $\frac{x^2 - 7x + 21}{x - 4}$ can be written in the form $x - a + \frac{b}{x - c}$, where a, b, and c are integers. what are the values of a, b, and c?
a= type your answer... b= type your answer... c= type your answer...

Explanation:

Step1: Polynomial long division setup

We divide $x^2 -7x +21$ by $x-4$. First, divide the leading term of the numerator by the leading term of the denominator: $\frac{x^2}{x}=x$.

Step2: Multiply divisor by result

Multiply $x-4$ by $x$: $x(x-4)=x^2-4x$.

Step3: Subtract from numerator

Subtract this from the original numerator:
$$(x^2-7x+21)-(x^2-4x)=-3x+21$$

Step4: Divide new leading term

Divide $-3x$ by $x$: $\frac{-3x}{x}=-3$.

Step5: Multiply divisor by new result

Multiply $x-4$ by $-3$: $-3(x-4)=-3x+12$.

Step6: Subtract to find remainder

Subtract this from $-3x+21$:
$$(-3x+21)-(-3x+12)=9$$

Step7: Rewrite original expression

Combine the terms: $\frac{x^2-7x+21}{x-4}=x-3+\frac{9}{x-4}$

Answer:

$a= -3$, $b=9$, $c=4$