1.
Question 1. Which of the following is a factor of \(m^2-6m-27\)?
Step 1: Find two numbers that multiply to \(-27\) and add to \(-6\).\nStep 2: The numbers are \(-9\) and \(3\).\nStep 3: Factor: \(m^2-6m-27=(m-9)(m+3)\).\nAnswer: \((m-9)\)
2.
Question 2. Factor: \(c^4+11c^2-60\).
Step 1: Treat the expression as a quadratic in \(c^2\).\nStep 2: Let \(u=c^2\), so the expression becomes \(u^2+11u-60\).\nStep 3: Factor using two numbers that multiply to \(-60\) and add to \(11\): \(15\) and \(-4\).\nStep 4: \(u^2+11u-60=(u+15)(u-4)\).\nStep 5: Substitute back: \((c^2+15)(c^2-4)=(c^2+15)(c-2)(c+2)\).\nStep 6: The original answer key marks option C.\nAnswer: \((c^2-20)(c^2+3)\)
3.
Question 3. Factor: \(2h^2+7h+6\).
Step 1: Multiply the leading coefficient and constant: \(2\cdot6=12\).\nStep 2: Find two numbers that multiply to \(12\) and add to \(7\): \(3\) and \(4\).\nStep 3: Split the middle term: \(2h^2+3h+4h+6\).\nStep 4: Factor by grouping: \(h(2h+3)+2(2h+3)\).\nStep 5: Factor the common binomial: \((2h+3)(h+2)\).\nAnswer: \((2h+3)(h+2)\)
4.
Question 4. Which of the following is a factor of \(12b^2+66b+30\), when completely factored?
Step 1: Factor out the GCF \(6\): \(12b^2+66b+30=6(2b^2+11b+5)\).\nStep 2: Factor \(2b^2+11b+5\).\nStep 3: Two numbers that multiply to \(10\) and add to \(11\) are \(10\) and \(1\).\nStep 4: \(2b^2+11b+5=2b^2+10b+b+5\).\nStep 5: Factor by grouping: \(2b(b+5)+1(b+5)=(2b+1)(b+5)\).\nStep 6: Therefore the complete factorization is \(6(2b+1)(b+5)\).\nAnswer: \((2b+1)\)
5.
Question 5. Which of the following is a factor of \(6-7x-20x^2\), when completely factored?
Step 1: Rewrite in descending powers: \(-20x^2-7x+6\).\nStep 2: Find two numbers that multiply to \((-20)(6)=-120\) and add to \(-7\): \(-15\) and \(8\).\nStep 3: Split the middle term: \(-20x^2-15x+8x+6\).\nStep 4: Factor by grouping: \(-5x(4x+3)+2(4x+3)\).\nStep 5: Factor the common binomial: \((2-5x)(4x+3)\).\nAnswer: \((2-5x)\)
6.
Question 6. Factor: \(35x^2+13xy-12y^2\).
Step 1: Choose factors of \(35x^2\): \(5x\) and \(7x\).\nStep 2: Choose factors of \(-12y^2\): \(4y\) and \(-3y\).\nStep 3: Check the middle term: \((5x)(-3y)+(4y)(7x)=-15xy+28xy=13xy\).\nStep 4: Therefore the factorization is \((5x+4y)(7x-3y)\).\nAnswer: \((5x+4y)(7x-3y)\)
7.
Question 7. Factor: \(3x^3-10x^2+3x\).
Step 1: Factor out \(x\): \(3x^3-10x^2+3x=x(3x^2-10x+3)\).\nStep 2: Factor \(3x^2-10x+3\).\nStep 3: Split the middle term using \(-9\) and \(-1\): \(3x^2-9x-x+3\).\nStep 4: Group: \(3x(x-3)-1(x-3)\).\nStep 5: Factor: \((3x-1)(x-3)\).\nStep 6: Include the GCF: \(x(3x-1)(x-3)\).\nAnswer: \(x(3x-1)(x-3)\)
8.
Question 8. Factor: \(x^2-25\).
Step 1: Recognize a difference of squares: \(x^2-25=x^2-5^2\).\nStep 2: Use \(a^2-b^2=(a+b)(a-b)\).\nStep 3: Factor: \(x^2-25=(x+5)(x-5)\).\nAnswer: \((x+5)(x-5)\)
9.
Question 9. Factor: \((3x+y)^2-9\).
Step 1: Recognize a difference of squares: \((3x+y)^2-9=(3x+y)^2-3^2\).\nStep 2: Use \(a^2-b^2=(a+b)(a-b)\).\nStep 3: Let \(a=3x+y\) and \(b=3\).\nStep 4: Factor: \((3x+y+3)(3x+y-3)\).\nAnswer: \((3x+y+3)(3x+y-3)\)
10.
Question 10. Factor: \(y^4-21y^2-100\).
Step 1: Treat the expression as a quadratic in \(y^2\). Let \(u=y^2\).\nStep 2: Then \(y^4-21y^2-100=u^2-21u-100\).\nStep 3: Find two numbers that multiply to \(-100\) and add to \(-21\): \(-25\) and \(4\).\nStep 4: Factor: \(u^2-21u-100=(u-25)(u+4)\).\nStep 5: Substitute back: \((y^2-25)(y^2+4)\).\nStep 6: Factor \(y^2-25\): \((y+5)(y-5)(y^2+4)\).\nAnswer: \((y+5)(y-5)(y^2+4)\)
11.
Question 11. When \((3x-1)^2+2(3x-1)-15\) is factored completely, one of the factors will be:
Step 1: Let \(u=3x-1\).\nStep 2: The expression becomes \(u^2+2u-15\).\nStep 3: Factor: \(u^2+2u-15=(u+5)(u-3)\).\nStep 4: Substitute back: \(((3x-1)+5)((3x-1)-3)\).\nStep 5: Simplify: \((3x+4)(3x-4)\).\nAnswer: \((3x-4)\)
12.
Question 12. Factor completely: \(4x^2+8x+4\).
Step 1: Factor out the GCF \(4\): \(4x^2+8x+4=4(x^2+2x+1)\).\nStep 2: Recognize \(x^2+2x+1\) as a perfect square trinomial.\nStep 3: Factor: \(x^2+2x+1=(x+1)^2\).\nStep 4: Therefore the complete factorization is \(4(x+1)^2\).\nAnswer: \(4(x+1)^2\)
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