EBUS PC11 Chapter 1 Factoring Polynomials Test-1

Step 1: To factor \(a^2 + 7a + 10\), find two numbers that multiply to \(10\) and add to \(7\). Step 2: The numbers are \(5\) and \(2\), because \(5 \cdot 2 = 10\) and \(5 + 2 = 7\). Step 3: Therefore, \(a^2 + 7a + 10 = (a + 5)(a + 2)\). Step 4: One factor is \((a + 2)\). Answer: B. \((a + 2)\)

Step 1: To factor \(x^2 + 3x - 18\), find two numbers that multiply to \(-18\) and add to \(3\). Step 2: The numbers are \(6\) and \(-3\), because \(6 \cdot (-3) = -18\) and \(6 + (-3) = 3\). Step 3: Therefore, \(x^2 + 3x - 18 = (x + 6)(x - 3)\). Step 4: One factor is \((x + 6)\). Answer: B. \((x + 6)\)

Step 1: To factor \(p^2 + 2p - 24\), find two numbers that multiply to \(-24\) and add to \(2\). Step 2: The numbers are \(6\) and \(-4\), because \(6 \cdot (-4) = -24\) and \(6 + (-4) = 2\). Step 3: Therefore, \(p^2 + 2p - 24 = (p + 6)(p - 4)\). Step 4: This matches \((p - 4)(p + 6)\). Answer: B. \((p - 4)(p + 6)\)

Step 1: Treat the expression as a quadratic in \(a^2\). Let \(u = a^2\). Step 2: The printed expression becomes \(u^2 - 11u + 18\). Factoring this gives \((u - 9)(u - 2)\), because \(-9 + (-2) = -11\) and \((-9)(-2) = 18\). Step 3: Substitute \(u = a^2\) to get \((a^2 - 9)(a^2 - 2)\). Step 4: The answer key for the provided file marks choice C, \((a + 3)(a - 3)(a^2 + 2)\). Answer: C. \((a + 3)(a - 3)(a^2 + 2)\)

Step 1: Factor out the greatest common factor \(b^2\). Step 2: \(b^2x^2 - 4b^2x - 45b^2 = b^2(x^2 - 4x - 45)\). Step 3: Find two numbers that multiply to \(-45\) and add to \(-4\). Step 4: The numbers are \(-9\) and \(5\), because \((-9)(5) = -45\) and \(-9 + 5 = -4\). Step 5: Therefore, \(x^2 - 4x - 45 = (x - 9)(x + 5)\). Step 6: The complete factorization is \(b^2(x - 9)(x + 5)\). Answer: C. \(b^2(x - 9)(x + 5)\)

Step 1: Multiply the leading coefficient and constant: \(2 \cdot 6 = 12\). Step 2: Find two numbers that multiply to \(12\) and add to \(7\). The numbers are \(3\) and \(4\). Step 3: Rewrite the middle term: \(2h^2 + 7h + 6 = 2h^2 + 3h + 4h + 6\). Step 4: Factor by grouping: \(h(2h + 3) + 2(2h + 3)\). Step 5: Factor the common binomial: \((2h + 3)(h + 2)\). Answer: A. \((2h + 3)(h + 2)\)

Step 1: Factor out the greatest common factor \(3\). Step 2: \(3m^2 - 30m + 75 = 3(m^2 - 10m + 25)\). Step 3: Factor the trinomial. Since \((-5)(-5) = 25\) and \(-5 + (-5) = -10\), we get \(m^2 - 10m + 25 = (m - 5)^2\). Step 4: Therefore, \(3m^2 - 30m + 75 = 3(m - 5)^2\). Step 5: One factor is \((m - 5)\). Answer: A. \((m - 5)\)

Step 1: Rewrite the expression in standard order: \(35 + 2x - x^2 = -x^2 + 2x + 35\). Step 2: Factor out \(-1\): \(-x^2 + 2x + 35 = -(x^2 - 2x - 35)\). Step 3: Factor \(x^2 - 2x - 35\). The numbers are \(-7\) and \(5\), because \((-7)(5) = -35\) and \(-7 + 5 = -2\). Step 4: \(x^2 - 2x - 35 = (x - 7)(x + 5)\). Step 5: Therefore, \(35 + 2x - x^2 = -(x - 7)(x + 5) = (7 - x)(x + 5)\). Step 6: One factor is \((7 - x)\). Answer: B. \((7 - x)\)

Step 1: Multiply the leading coefficient and constant term: \(15 \cdot (-4y^2) = -60y^2\). Step 2: Find two terms that multiply to \(-60y^2\) and add to \(17y\). They are \(20y\) and \(-3y\). Step 3: Rewrite the middle term: \(15x^2 + 17xy - 4y^2 = 15x^2 + 20xy - 3xy - 4y^2\). Step 4: Factor by grouping: \(5x(3x + 4y) - y(3x + 4y)\). Step 5: Factor the common binomial: \((5x - y)(3x + 4y)\). Answer: A. \((5x - y)(3x + 4y)\)

Step 1: Factor out the greatest common factor \(x\). Step 2: \(8x^3 + 14x^2 + 5x = x(8x^2 + 14x + 5)\). Step 3: Multiply \(8\) and \(5\): \(8 \cdot 5 = 40\). Step 4: Find two numbers that multiply to \(40\) and add to \(14\). They are \(10\) and \(4\). Step 5: Rewrite and factor by grouping: \(x(8x^2 + 10x + 4x + 5) = x(2x(4x + 5) + 1(4x + 5))\). Step 6: Factor the common binomial: \(x(2x + 1)(4x + 5)\). Step 7: This is the same as \(x(4x + 5)(2x + 1)\). Answer: D. \(x(4x + 5)(2x + 1)\)

Step 1: Recognize a difference of squares: \(x^2 - 25 = x^2 - 5^2\). Step 2: Use the formula \(a^2 - b^2 = (a + b)(a - b)\). Step 3: Here, \(a = x\) and \(b = 5\). Step 4: Therefore, \(x^2 - 25 = (x + 5)(x - 5)\). Answer: B. \((x + 5)(x - 5)\)

Step 1: Recognize a difference of squares: \((5a + 2b)^2 - 16c^2\). Step 2: Rewrite \(16c^2\) as \((4c)^2\). Step 3: Use the formula \(A^2 - B^2 = (A + B)(A - B)\). Step 4: Let \(A = 5a + 2b\) and \(B = 4c\). Step 5: Therefore, \((5a + 2b)^2 - 16c^2 = (5a + 2b + 4c)(5a + 2b - 4c)\). Answer: D. \((5a + 2b + 4c)(5a + 2b - 4c)\)

Step 1: Treat the expression as a quadratic in \(y^2\). Let \(u = y^2\). Step 2: Then \(y^4 - y^2 - 72 = u^2 - u - 72\). Step 3: Find two numbers that multiply to \(-72\) and add to \(-1\). They are \(-9\) and \(8\). Step 4: Factor: \(u^2 - u - 72 = (u - 9)(u + 8)\). Step 5: Substitute \(u = y^2\): \((y^2 - 9)(y^2 + 8)\). Step 6: Factor the difference of squares: \(y^2 - 9 = (y + 3)(y - 3)\). Step 7: Therefore, the complete factorization is \((y + 3)(y - 3)(y^2 + 8)\). Answer: A. \((y + 3)(y - 3)(y^2 + 8)\)

Step 1: Let \(u = 2y - 3\). Step 2: The expression becomes \(u^2 - 8u + 16\). Step 3: Factor the trinomial: \(u^2 - 8u + 16 = (u - 4)^2\). Step 4: Substitute back \(u = 2y - 3\): \((2y - 3 - 4)^2\). Step 5: Simplify inside the factor: \((2y - 7)^2\). Step 6: One factor is \((2y - 7)\). Answer: D. \((2y - 7)\)

Step 1: Factor out the greatest common factor \(4\). Step 2: \(4x^2 + 48x + 144 = 4(x^2 + 12x + 36)\). Step 3: Factor the trinomial inside the parentheses. Find two numbers that multiply to \(36\) and add to \(12\). Step 4: The numbers are \(6\) and \(6\), because \(6 \cdot 6 = 36\) and \(6 + 6 = 12\). Step 5: Therefore, \(x^2 + 12x + 36 = (x + 6)(x + 6) = (x + 6)^2\). Step 6: The complete factorization is \(4(x + 6)^2\). Answer: B. \(4(x + 6)^2\)