#corresponding coefficients

[pwnthesat]

If (ax+b)(cx+d) = 6x^2+11x-10 for all values of x,what is the value of ac-bd? A)16 B)-4 C)-10 D)-11 E)-16

FOIL this out, and then remember that corresponding coefficients of equivalent polynomials are equal.

acx^2 + adx + bcx + bd = 6x^2 + 11x – 10

You can see that the coefficients of the x^2 terms are ac and 6, so you know that ac = 6. Likewise, you can see that the constant terms are bd and 10, so you know that bd = –10. The question asks for ac – bd, so that’s 6 – (–10) = 16.

[pwnthesat]

If a(x+1)+b(x-1)=7x for all real number x, where a and b are constants, what is the value of a?

Do a little rearranging: 

ax + a + bx – b = 7x(a + b)x + (a – b) = 7x + 0

I added that little “+0″ at the end to drive home the fact that we’re dealing with equivalent polynomials. Because we know that the coefficients of equivalent polynomials are equal, we know that a + b = 7 and a – b = 0. So we can solve for a by adding the two equations:

a + b + a – b = 7 + 02a = 7a = 3.5

[pwnthesat]

Hello sorry im bothering you but theres a question that i dont get it in page 314 18 you wrote the answer as 12 but if we set the formula as 12 and 1 for b it will not work cause we have square of x and 12x plus 1x and 12 not 40 as c so it willnt equal each other so please would you explain it

All that’s going on here is that you know that x^2 + 13x + 40 is equivalent to x^2 + ax + bx + 40. The constants a and b can be ANYTHING as long as their sum is 13. They could be 5 and 8, 6 and 7, 2 and 11, 1 and 12, etc.

Forget that you got x^2 + 13x + 40 by FOILing (x + 5)(x + 8). Once you’ve got x^2 + 13x + 40, it doesn’t matter where it came from; a and b don’t have to be 5 and 8.

[pwnthesat]

If x^2+px+r = (x+7)(x+4) for all values of x, where p and r are constants, what is the value of pr ?

This is a question about equivalent polynomials, which I have a post on here.

If you FOIL out the right-hand side, you get x^2 + 11x + 28. That tells you that p = 11 and r = 28. Therefore, pr = 308.

[pwnthesat]

X^2-9x+18= X(X-3a)+(ab). The equation above is true for all values of x. If a and b are constants, what is the value of b ? (3) (6) (9) (12) (18)

This is all about corresponding coefficients.

x^2 – 9x + 18 = x^2 – 3ax + ab

That tells you that 9 = 3a, and 18 = ab. 

If 9 = 3a, then a = 3. 

If 18 = ab and a = 3, then b = 6.

[pwnthesat]

X^2(3X-1)=a+bx+sx^2+tx^3. This equation is true for all values of X, where a,b,s, and t are constant. what is the value of a+b+s+t? (-2) (1) (2) (3) (4)

When you simplify that, you get 3x^3 – x^2 = a + bx + sx^2 + tx^3. That tells you that a = 0, b = 0, s = –1, and t = 3. So a + b + s + t = 2.

What the heck is going on here? Corresponding coefficients, that’s what.

[pwnthesat]

x^2(3x-1)= a+bx+sx^2+tx^3. The equation above is true for all values of x, where a,b,s, and t are constants. What is the value of a+b+s+t? A) -2 B) 1 C)2 D) 3 E) 4 Thanks Mike!!

You should read this post.

3x^3 – x^2 = a + bx + sx^2 + tx^3

That tells you that a and b equal 0, s = –1, and t = 3. So a + b + s + t = 2. 

[pwnthesat]

If 90(x^3 + 1/10x^2 +1/30x +1/90) = x^3 + bx^2 + cx +d for all values of x , where a,b,c,d are constants, what is the value of a+b+c+d ?

I think you forgot to type an a—that should be the coefficient of the x^3 term on the right hand side, yes? PLEASE type carefully.

Anyway, this is all about corresponding coefficients. Distribute that 90 and you get:

90x^3 + 9x^2 + 3x + 1 = ax^3 + bx^2 + cx + d

That means a = 90, b = 9, c = 3, and d = 1. Add those up:

90 + 9 + 3 + 1 = 103