Factoring Polynomial Expressions: A Comprehensive ApproachFactor the following expression completely. If the polynomial is prime, then state this as youranswer.49x square root–84xy + 36y square root = (7x–6y)²Answer:Let's go step bystep to factor the given expression:Expression:49x2−84xy+36y249x^2-84xy + 36y^2Step 1: Recognize the structureThe given expression is a quadratic trinomial. We can try to factor it by recognizing it as aperfect square trinomial.A perfectsquare trinomial takes the form:a2−2ab+b2=(a−b)2a^2-2ab + b^2 = (a-b)^2Where:•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 formWe can try to rewrite49x2−84xy+36y249x^2-84xy + 36y^2 to see if it fits the perfectsquare 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 trinomialSince we have:49x2−84xy+36y2=(7x−6y)249x^2-84xy + 36y^2 = (7x-6y)^2Final Answer:The factored form of 49x2−84xy+36y249x^2-84xy + 36y^2 is:(7x−6y)2(7x-6y)^2Factor the following expression completely. If the polynomial is prime, then state this as youranswer.64x tothe third–1 = (4x-1)(16x² +4x +1)Answer:Let's factor the given expression:Expression:64x3−164x^3-1Step 1: Recognize the patternThis is adifference 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)Preview Mode
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