Question 19 of 43intermediate🔧 ApplyFill in the Blanks4 marks

Fill in the blanks to complete the following identities:

(i)

s2s^{2} - 11s + 24 = ()()

Answer

s - 3, s - 8

(ii)

(____)(x + 1) = (3x23x^{2} - 4x - 7)

Answer

3x - 7

(iii)

10x210x^{2} - 11x - 6 = (2x - )( + 2)

Answer

3, 5x

(iv)

6x26x^{2} + 7x + 2 = ()()

Answer

2x + 1, 3x + 2

Explanation

Using the identities (x+a)(x+b)=x2+(a+b)x+ab(x+a)(x+b) = x^2 + (a+b)x + ab and (ax+b)(cx+d)=acx2+(ad+bc)x+bd(ax+b)(cx+d) = acx^2 + (ad+bc)x + bd provided in the chapter, we can find the missing factors by comparing coefficients. For (i), we need two numbers that add to -11 and multiply to 24, which are -3 and -8. For (ii), expanding (ax+b)(x+1)(ax+b)(x+1) and comparing to 3x2−4x−73x^2 - 4x - 7 gives a=3,b=−7a=3, b=-7. For (iii), expanding (2x−a)(bx+2)(2x-a)(bx+2) and comparing to 10x2−11x−610x^2 - 11x - 6 gives a=3,b=5a=3, b=5. For (iv), finding factors of 6 and 2 that give a cross-sum of 7 yields (2x+1)(2x+1) and (3x+2)(3x+2).