Polynomials

Hong Kong

Stage 4 - Stage 5

The polynomials $4x^2-7x-15$4`x`2−7`x`−15 and $5x^2+13x+k$5`x`2+13`x`+`k` have a common factor of $x+p$`x`+`p`, where $p$`p` is an integer.

a

Using the fact that $x+p$`x`+`p` is a factor of $4x^2-7x-15$4`x`2−7`x`−15, solve for the value of $p$`p`.

b

Using the fact that $x+p$`x`+`p` is a factor of $5x^2+13x+k$5`x`2+13`x`+`k`, solve for $k$`k`.

Easy

Approx 7 minutes

Consider the polynomial $x^{99}+1$`x`99+1.

The polynomials $P\left(x\right)=x^3+4x^2-5x+n$`P`(`x`)=`x`3+4`x`2−5`x`+`n` and $Q\left(x\right)=x^3+2x+17$`Q`(`x`)=`x`3+2`x`+17 leave the same remainder when divided by $x+1$`x`+1.

Solve for the value of $n$`n`.

Consider the sequence: $2^5-2$25−2, $3^5-3$35−3, $4^5-4$45−4, $\text{. . .}$. . ., $x^5-x$`x`5−`x`.

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