AP DSC School Assistant Math Model Test 2

Concept:This question uses the algebraic identities for perfect squares and difference of squares to factorize a given expression into a product of two linear factors.Explanation:We start with the expression:$x^2 - y^2 - 9z^2 + 6yz$Step 1: Group the terms that involve $y$ and $z$ and factor out a minus sign:$x^2 - (y^2 + 9z^2 - 6yz)$Step 2: Recognize that $(y^2 + 9z^2 - 6yz)$ is a perfect square. Use the identity $(a-b)^2 = a^2 + b^2 - 2ab$ with $a = y$ and $b = 3z$:$y^2 + (3z)^2 - 2(y)(3z) = (y - 3z)^2$So the expression becomes:$x^2 - (y - 3z)^2$Step 3: Now apply the difference of squares identity $a^2 - b^2 = (a - b)(a + b)$ with $a = x$ and $b = (y - 3z)$:$(x - (y - 3z))(x + (y - 3z))$Step 4: Simplify the brackets:$(x - y + 3z)(x + y - 3z)$Thus the factorization is $(x - y + 3z)(x + y - 3z)$.Answer:Option C: $(x - y + 3z)(x + y - 3z)$.