The polynomial which results from the expansion of \((x^2+5x+6)^2+(px+q)(x^3+7x^2+3x)\) has degree 2. Find p+q.
THX in advance
The degree of a function is the exponent of the leading term. For example: in the polynomial x^2 + 3x + 4, the degree is 2 because it's the exponent of the leading term.
+128407 (x^2 + 5x + 6)^2 =
(x^2 +5x + 6) (x^2 + 5x + 6) =
x^4 +5x^3 + 6x^2
+ + 5x^3 +25x^2 + 30x
+ 6x^2 + 30x + 36
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x^4 + 10x^3 + 37x^2 + 60x + 36 (1)
And
(px + q) (x^3 + 7x^2 + 3x) =
px^4 + 7px^3 + 3px^2
+ qx^3 + 7qx^2 + 3qx
__________________________
px^4 + (7p+q)x^3 + (3p + 7q)x^2 + 3qx (2)
Combining these we have
(1 + p)x^4 + (10 + 7p + q)x^3 + (37 + 3p + 7q)x^2 + (60+ 3q)x + 36
Letting p = -1 we can eliminate the x^4 term
(1 - 1)x^4 + ( 10 + 7(-1) + q)x^3 + (37 + 3(-1) + 7q)x^2 + (60 + 3q)x + 36 =
(3 + q)x^3 + ( 34 + 7q)x^2 + (60 + 3q)x+ 36
Letting q =-3 and we can eliminate the x^3 term
(3 + -3)x^3 + (34 + 7(-3) )x^2 + (60 + 3(-3) )x + 36 =
13x^2 + 51x + 36 = degree 2
And p +q = -1 + (-3) = -4
The degree of a polynomial function[in one variable] is determined the highest power on that variable
