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Factoring Polynomial Expressions: A Comprehensive Approach

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    Factoring Polynomial Expressions: A Comprehensive Approach Factor the following expression completely. If the polynomial is prime, then state this as your answer. 49x square root – 84xy + 36y square root = (7x – 6y)² Answer: Let's go step by step to factor the given expression: Expression: 49x2−84xy+36y249x^2 - 84xy + 36y^2 Step 1: Recognize the structure The given expression is a quadratic trinomial. We can try to factor it by recognizing it as a perfect square trinomial. A perfect square trinomial takes the form: a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2 Where: • a2a^2 is the first term, • 2ab2ab is the middle term, • b2b^2 is the last term. Step 2: Compare terms with the perfect square trinomial form We can try to rewrite 49x2−84xy+36y249x^2 - 84xy + 36y^2 to see if it fits the perfect square trinomial form: • The first term is 49x249x^2, which is the square of 7x7x (since (7x)2=49x2(7x)^2 = 49x^2). • The last term is 36y236y^2, which is the square of 6y6y (since (6y)2=36y2(6y)^2 = 36y^2). • The middle term is −84xy - 84xy, which is −2×7x×6y - 2 \ times 7x \ times 6y, or −84xy - 84xy. So, the expression does indeed match the form of a perfect square trinomial. Step 3: Factor the perfect square trinomial Since we have: 49x2−84xy+36y2=(7x−6y)249x^2 - 84xy + 36y^2 = (7x - 6y)^2 Final Answer: The factored form of 49x2−84xy+36y249x^2 - 84xy + 36y^2 is: (7x−6y)2(7x - 6y)^2 Factor the following expression completely. If the polynomial is prime, then state this as your answer. 64x to the third – 1 = (4x - 1)(16x² +4x +1) Answer: Let's factor the given expression: Expression: 64x3−164x^3 - 1 Step 1: Recognize the pattern This is a difference of cubes , since it fits the formula: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2) In this case: • a=4xa = 4x (since (4x)3=64x3(4x)^3 = 64x^3)

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    • b=1b = 1 (since 13=11^3 = 1) Step 2: Apply the difference of cubes formula Substitute a=4xa = 4x and b=1b = 1 into the formula: 64x3−1=(4x−1)((4x)2+(4x)(1)+12)64x^3 - 1 = (4x - 1) \ left((4x)^2 + (4x)(1) + 1^2 \ right) Step 3: Simplify the terms inside the parentheses Now, let's simplify each term inside the second factor: (4x)2=16x2(4x)^2 = 16x^2 (4x)(1)=4x(4x)(1) = 4x 12=11^2 = 1 So the factored expression becomes: 64x3−1=(4x−1)(16x2+4x+1)64x^3 - 1 = (4x - 1)(16x^2 + 4x + 1) Final Answer: The completely factored form of 64x3−164x^3 - 1 is: (4x−1)(16x2+4x+1)(4x - 1)(16x^2 + 4x + 1) Since the second factor, 16x2+4x+116x^2 + 4x + 1, cannot be factored further (it is a quadratic with no real roots), this is the final factored form. Factor the following expression completely, if the polynomial is prime, then state this as your answer. 36y square root + 60y + 25 = (6x+5)² Answer: Let's examine and factor the expression: Expression: 36y2+60y+2536y^2 + 60y + 25 Step 1: Recognize the pattern This expression is a perfect square trinomial . A perfect square trinomial follows the form: a2+2ab+b2=(a+b )2a^2 + 2ab + b^2 = (a + b)^2 We need to identify aa and bb such that: a2=36y2andb2=25a^2 = 36y^2 \ quad \ text{and} \ quad b^2 = 25 Step 2: Find aa and bb • For a2=36y2a^2 = 36y^2, we take the square root of 36y², which gives a=6ya = 6y. • For b2=25b^2 = 25, we take the square root of 25, which gives b=5b = 5. Step 3: Check the middle term The middle term of a perfect square trinomial is 2ab2ab. Let’s check if it matches the middle term in our expression: 2ab=2(6y)(5)=60y2ab = 2(6y)(5) = 60y This matches the middle term in the original expression. Step 4: Write the factored form Since it is a perfect square trinomial, the expression can be factored as: 36y2+60y+25=(6y+5)236y^2 + 60y + 25 = (6y + 5)^2 Final Answer: The completely factored form is: (6y+5)2(6y + 5)^2 Use factoring techniques to solve the following quadratic equation. z square root + z – 2 = 0 z = 1 or z = - 2

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    Answer: Let's solve the quadratic equation step by step: Given equation: z2+z−2=0z^2 + z - 2 = 0 Step 1: Factor the quadratic expression We need to factor the quadratic expression z2+z−2z^2 + z - 2. We are looking for two numbers that multiply to −2 - 2 (the constant term) and add up to 11 (the coefficient of zz). The two numbers are 22 and −1 - 1, because: • 2×−1=−22 \ times - 1 = - 2 • 2+(−1)=12 + ( - 1) = 1 So we can factor the quadratic as: z2+z−2=(z+2)(z−1)z^2 + z - 2 = (z + 2)(z - 1) Step 2: Set each factor equal to 0 Now that we have factored the equation, we set each factor equal to 0: (z+2)=0or(z−1)=0(z + 2) = 0 \ quad \ text{or} \ quad (z - 1) = 0 Step 3: Solve for zz 1. From z+2=0z + 2 = 0, we get: z=−2z = - 2 2. From z−1=0z - 1 = 0, we get: z=1z = 1 Final Answer: The solutions to the equation are: z=1orz=−2z = 1 \ quad \ text{or} \ quad z = - 2 Factor the following expression completely, if the polynomial is prime, then state this as your answer. 27x to the third + 8y to the third = (3x + 2y)(9x² - 6xy + 4y²) Answer: Let's factor the given expression step by step. Given expression: 27x3+8y327x^3 + 8y^3 This is a sum of cubes, which can be factored using the sum of cubes formula : a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) Step 1: Identify aa and bb In this case, we can rewrite 27x327x^3 and 8y38y^3 as cubes: • 27x3=(3x)327x^3 = (3x)^3 • 8y3=(2y)38y^3 = (2y)^3 So, a=3xa = 3x and b=2yb = 2y. Step 2: Apply the sum of cubes formula Using the sum of cubes formula: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) Substituting a=3xa = 3x and b=2yb = 2y: 27x3+8y3=(3x+2y)((3x)2−(3x)(2y)+(2y)2)27x^3 + 8y^3 = (3x + 2y)((3x)^2 - (3x)(2y) + (2y)^2) Step 3: Simplify the terms inside the parentheses Now, simplify each term in the second factor: • (3x)2=9x2(3x)^2 = 9x^2 • (3x)(2y)=6xy(3x)(2y) = 6xy • (2y)2=4y2(2y)^2 = 4y^2 So, the factorization becomes: 27x3+8y3=(3x+2y)(9x2−6xy+4y2)27x^3 + 8y^3 = (3x + 2y)(9x^2 - 6xy + 4y^2)

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