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QuestionFactor these expressions completely:
1. y2+8y+12y^{2}+8y+12
2. z2−z−90z^{2}-z-90
3. z2−9z+20z^{2}-9z+20
4. 4u3+4u2−360m4u^{3}+4u^{2}-360m

Studdy Solution

STEP 1

Assumptions1. We are given four expressions to factor completely. . The expressions are - y+8y+12y^{}+8 y+12 - z−z−90z^{}-z-90 - z−9z+20z^{}-9 z+20 - 4u3+4u−360m4 u^{3}+4 u^{}-360 m
3. We need to find two numbers that multiply to the constant term and add to the coefficient of the middle term.

STEP 2

Start with the first expression y2+8y+12y^{2}+8 y+12. We need to find two numbers that multiply to12 and add to8.

STEP 3

The two numbers that satisfy these conditions are2 and6. Therefore, we can write the expression asy2+8y+12=(y+2)(y+6)y^{2}+8 y+12 = (y+2)(y+6)

STEP 4

Next, consider the second expression z2−z−90z^{2}-z-90. We need to find two numbers that multiply to -90 and add to -1.

STEP 5

The two numbers that satisfy these conditions are -10 and9. Therefore, we can write the expression asz2−z−90=(z−10)(z+9)z^{2}-z-90 = (z-10)(z+9)

STEP 6

Now, consider the third expression z2−9z+20z^{2}-9 z+20. We need to find two numbers that multiply to20 and add to -9.

STEP 7

The two numbers that satisfy these conditions are -4 and -5. Therefore, we can write the expression asz2−9z+20=(z−4)(z−5)z^{2}-9 z+20 = (z-4)(z-5)

STEP 8

Finally, consider the fourth expression 4u3+4u2−360m4 u^{3}+4 u^{2}-360 m. This is a bit more complex because it involves three terms. First, factor out the common factor of44u3+4u2−360m=4(u3+u2−90m)4 u^{3}+4 u^{2}-360 m =4(u^{3}+u^{2}-90m)

STEP 9

Now, we need to factor the expression inside the parentheses. This is a cubic expression, and factoring it might be complex. It seems like there might be a mistake in the problem, as u3+u2−90mu^{3}+u^{2}-90m is not easily factorable. Let's assume it's a typo and the correct expression is u3+u2−90uu^{3}+u^{2}-90u.

STEP 10

Now, we need to find two numbers that multiply to -90 and add to. The two numbers that satisfy these conditions are -9 and10. Therefore, we can write the expression as4(u3+u2−90u)=4(u−9)(u+10)4(u^{3}+u^{2}-90u) =4(u-9)(u+10)So, the factored forms of the given expressions are. y2+8y+12=(y+2)(y+6)y^{2}+8 y+12 = (y+2)(y+6)2. z2−z−90=(z−10)(z+9)z^{2}-z-90 = (z-10)(z+9)3. z2−9z+20=(z−4)(z−5)z^{2}-9 z+20 = (z-4)(z-5)4. 4u3+4u2−360m=4(u−9)(u+10)4 u^{3}+4 u^{2}-360 m =4(u-9)(u+10) (assuming a typo in the original problem)

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