l-b-and-mn-must-equal-c-in-x2-bx-c-which-of-the-following-derivations-is-correct

Question

Explain, using a short algebraic derivation starting from: (x+m)(x+n)(x + m)(x + n), why m+nm + n must equal bb and m⋅nm\cdot n must equal cc in: $$x2+bx+cx^2 + bx + c. $$Which of the following derivations shown below is correct?

Accepted Answer

Expand (x+m)(x+n)=x2+nx+mx+mn=x2+(m+n)x+mn(x + m)(x + n) = $$x^2 + nx + mx + mn$$ = $$x^2 + (m + n)x + mn$$, so comparing coefficients gives m+n=bm + n = b and mn=cmn = c

Answer

Expand (x+m)(x+n)=x2+m2+n2+mn(x + m)(x + n) = $$x^2$$ + $$m^2$$ + $$n^2 + mn$$, and by equating coefficients we get m+n=bm + n = b and mn=cmn = c which follows from rearranging terms

Answer

Assume (x+m)(x+n)=x2+bx+c(x + m)(x + n) = $$x^2 + bx + c $$and substitute x=1x = 1 to obtain m+n=bm + n = b and x=0x = 0 to get mn=cmn = c so these must hold for all xx

Answer

Multiply termwise to get $$x2+m+n+mnx^2 + m + n + mn $$but since xx terms disappear we set m+n=bm + n = b and mn=cmn = c

Explain, using a short algebraic derivation starting from: (x+m)(x+n)(x + m)(x + n), why m+nm + n must equal bb and m⋅nm\(\cdot\) n must equal cc in: x2+bx+cx^2 + bx + c. Which of the following derivations shown below is correct?

Explain, using a short algebraic derivation starting from: $(x + m) (x + n)$ , why $m + n$ must equal $b$ and $m \cdot n$ must equal $c$ in: $x^{2} + b x + c$ . Which of the following derivations shown below is correct?