katehorton11
Answered

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Divide.

[tex]\[
\left(2x^2 + 17x + 36\right) \div (x + 5)
\][/tex]

Your answer should include both the quotient and the remainder.

Sagot :

GinnyAnswer
Sure, let's divide the polynomial [tex]\(2x^2 + 17x + 36\)[/tex] by [tex]\(x + 5\)[/tex]. We'll use the polynomial long division method.

### Step-by-Step Division:

1. Setup the division:
Write the dividend [tex]\(2x^2 + 17x + 36\)[/tex] and the divisor [tex]\(x + 5\)[/tex].

2. Divide the leading terms:
Divide the leading term of the dividend [tex]\(2x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex].

[tex]\[ \frac{2x^2}{x} = 2x \][/tex]

So, the first term of the quotient is [tex]\(2x\)[/tex].

3. Multiply and subtract:
Multiply the entire divisor [tex]\(x + 5\)[/tex] by the first term of the quotient [tex]\(2x\)[/tex]:

[tex]\[ (2x)(x + 5) = 2x^2 + 10x \][/tex]

Now, subtract this result from the original dividend:

[tex]\[ (2x^2 + 17x + 36) - (2x^2 + 10x) = 7x + 36 \][/tex]

4. Repeat the process:
Now, take the new dividend [tex]\(7x + 36\)[/tex] and divide its leading term [tex]\(7x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:

[tex]\[ \frac{7x}{x} = 7 \][/tex]

So, the next term of the quotient is [tex]\(7\)[/tex].

5. Multiply and subtract:
Multiply the entire divisor [tex]\(x + 5\)[/tex] by the new term of the quotient [tex]\(7\)[/tex]:

[tex]\[ (7)(x + 5) = 7x + 35 \][/tex]

Subtract this from the new dividend:

[tex]\[ (7x + 36) - (7x + 35) = 1 \][/tex]

### Results:
After completing the division, the quotient is [tex]\(2x + 7\)[/tex] and the remainder is [tex]\(1\)[/tex].

Thus, the division of [tex]\(2x^2 + 17x + 36\)[/tex] by [tex]\(x + 5\)[/tex] yields:
- Quotient: [tex]\(2x + 7\)[/tex]
- Remainder: [tex]\(1\)[/tex]

So, the final answer is:

[tex]\[ \boxed{(2x + 7, 1)} \][/tex]
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