<p>from practice test 3, section 5 in the blue book (newest edition):</p>
<p>(x-8)(x-k) = x^2 - 5kx + m</p>
<p>In the equation above, k and m are constants. If the equation is true for all values of x, what is the value of m?</p>
<p>(A) 8
(B) 16
(C) 24
(D) 32
(E) 40</p>
<p>Please show me how to do this... thanks!</p>
<p>Foil out the left and you get: x^2-kx-8x+8k=x^2-5kx+m</p>
<p>Cancel out the x^2 and add kx and you have: 4kx-8x+8k=m</p>
<p>At this point, I would try to get the m and k on opposite sides so when you plug something in for x, the k can cancel leaving the m alone.</p>
<p>So, 4kx+8k=8x+m</p>
<p>Now factor the left hand side</p>
<p>4k(x+2)=8x+m</p>
<p>At this point you want to plug in an x value that will give you 0 to cancel out k and leave m. If you plug in -2 for x…</p>
<p>4k(-2+2)=8(-2)+m
0=-16+m
m = 16 </p>
<p>which is answer choice B</p>
<p>I was directed to this post when I searched for “hard SAT math problems”.</p>
<p>While the answer is correct there is an underlying concept that important to understand. It could help you in solving similar problems quickly.</p>
<p>If you have 2 polynomial functions, f(x) and g(x), and they are equal for all values of x then all the coefficients of the powers of x are equal.</p>
<p>For example:</p>
<p>(1) linear functions: f(x) = Ax + B, g(x) = Cx + D, f(x)=g(x) for all x. Then A=C and B=D</p>
<p>(2) quadratic functions: Ax^2 + Bx + C = Dx^2 + Ex + F for all x. Then A=D, B=E, C=F</p>
<p>etc.</p>
<p>If you haven’t seen this before, it’s easy to prove. Substitute x = 0 to show that the constant coefficients are equal, and cancel out. Factor an x, and repeat …</p>
<p>For the specific problem expand the left hand side, to get: x^2 + (-k -8)x + 8k = x^2 + (-5k)x + m
So m = 8k and -5k = -k - 8. Solve the second equality and get k = 2, and m = 8k = 16.</p>
<p>Practice with similar problems.</p>
<p>^^ Thats the solution I would’ve done. Since x is constant and both equations are in the exact same form (quadratic), than each coefficient is equal.</p>