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(3y-2)(y+a)=3y^(2)+bz If the given equation is true for all values of y, where a and b are constants, which of the following is the value of b ? Choose 1 answer: (A) -38 (B) 12 (c) 34 (D) 36

(3y−2)(y+a)=3y2+bz (3 y-2)(y+a)=3 y^{2}+b z \newlineIf the given equation is true for all values of y y , where a a and b b are constants, which of the following is the value of b b ?\newlineChoose 11 answer:\newline(A) −38-38\newline(B) 1212\newline(C) 3434\newline(D) 3636

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Q. (3y−2)(y+a)=3y2+bz (3 y-2)(y+a)=3 y^{2}+b z \newlineIf the given equation is true for all values of y y , where a a and b b are constants, which of the following is the value of b b ?\newlineChoose 11 answer:\newline(A) −38-38\newline(B) 1212\newline(C) 3434\newline(D) 3636
  1. Expand and Compare: We need to expand the left side of the equation (3y−2)(y+a)(3y-2)(y+a) to compare it with the right side 3y2+bz3y^2+bz.(3y−2)(y+a)=3y(y)+3y(a)−2(y)−2(a)(3y-2)(y+a) = 3y(y) + 3y(a) - 2(y) - 2(a)
  2. Simplify Expression: Now we simplify the expression by multiplying the terms.\newline3y(y)+3y(a)−2(y)−2(a)=3y2+3ay−2y−2a3y(y) + 3y(a) - 2(y) - 2(a) = 3y^2 + 3ay - 2y - 2a
  3. Comparison with Right Side: We compare the simplified expression with the right side of the given equation.\newline3y2+3ay−2y−2a=3y2+bz3y^2 + 3ay - 2y - 2a = 3y^2 + bz
  4. Coefficient Comparison for yy: Since the equation is true for all values of yy, the coefficients of the corresponding terms on both sides must be equal. This means that the coefficient of yy on the left side (3a−2)(3a - 2) must be equal to the coefficient of yy on the right side, which is 00 (since there is no yy term on the right side).\newline3a−2=03a - 2 = 0
  5. Solve for aa: We solve for aa.3a−2=03a - 2 = 03a=23a = 2a=23a = \frac{2}{3}
  6. Constant Term Comparison: Now we look for the constant term on the left side, which is −2a-2a, and compare it with the constant term on the right side, which is bzbz. \newline−2a=bz-2a = bz
  7. Substitute Value of a: We substitute the value of aa into the equation.−2(23)=bz-2\left(\frac{2}{3}\right) = bz−43=bz-\frac{4}{3} = bz
  8. Determine Coefficient for bb: Since there is no zz term on the left side of the original equation, the coefficient of zz on the right side must be 00. Therefore, bb must be 00.
    bz=0bz = 0
    b=0b = 0

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