Polynomials

Hong Kong

Stage 4 - Stage 5

Lesson

Adding and subtracting polynomials may sound complicated but it's just the same process as simplifying algebraic expression and equations.

Remember!

Just like with any algebraic expression, we can only add and subtract like terms.

Let's run through the process by looking at an example. Let's say we want to find the difference between two polynomials: $P\left(x\right)=7x^3+4x^2-4$`P`(`x`)=7`x`3+4`x`2−4 and $Q\left(x\right)=7x^3+8x^2-2x-8$`Q`(`x`)=7`x`3+8`x`2−2`x`−8.

1. Start by writing out the equation we want to solve:

$P\left(x\right)-Q\left(x\right)=7x^3+4x^2-4-\left(7x^3+8x^2-2x-8\right)$`P`(`x`)−`Q`(`x`)=7`x`3+4`x`2−4−(7`x`3+8`x`2−2`x`−8)

2. Collect the like terms, taking any negative symbols into account. Remember if there is a term with no corresponding term in the other polynomial, we can treat this as a value of zero. For example, $P\left(x\right)$`P`(`x`) does not have a term with $x$`x` but $Q\left(x\right)$`Q`(`x`) does.

We can present it in a table

$7x^3$7x3 |
$+$+ | $4x^2$4x2 |
$+$+ | $0$0 | $-$− | $4$4 | |

$-$− | $7x^3$7x3 |
$+$+ | $8x^2$8x2 |
$-$− | $2x$2x |
$-$− | $8$8 |

$0$0 | $-$− | $4x^2$4x2 |
$+$+ | $2x$2x |
$+$+ | $4$4 |

or we can simply collect the like terms:

collecting the $x^3$x3's |
$7x^3-7x^3$7x3−7x3 |
$=$= | $0$0 |

collecting the $x^2$x2's |
$4x^2-8x^2$4x2−8x2 |
$=$= | $-4x^2$−4x2 |

collecting the $x$x's |
$0-\left(-2x\right)$0−(−2x) |
$=$= | $2x$2x |

collecting the constant terms | $-4-\left(-8\right)$−4−(−8) | $=$= | $4$4 |

You can choose what method you like.

3. Write out the solution

$P\left(x\right)-Q\left(x\right)=-4x^2+2x+4$`P`(`x`)−`Q`(`x`)=−4`x`2+2`x`+4

We may also be asked to describe features of the function resulting from the sum or difference of two polynomials, so it's important that we're familiar with the features of functions.

Here is a quick summary:

- A linear polynomial is the equation of a straight line, which can be written in the form $f\left(x\right)=mx+b$
`f`(`x`)=`m``x`+`b`. - A quadratic polynomial is the equation of a parabola, which can be written in the form $f\left(x\right)=ax^2+bx+c$
`f`(`x`)=`a``x`2+`b``x`+`c`. - A cubic polynomial, which can be written in the form $f\left(x\right)=ax^3+bx^2+cx+d$
`f`(`x`)=`a``x`3+`b``x`2+`c``x`+`d`. - A quartic polynomial, which can be written in the form $f\left(x\right)=ax^4+bx^3+cx^2+dx+e$
`f`(`x`)=`a``x`4+`b``x`3+`c``x`2+`d``x`+`e`.

Simplify $\left(3x^3-9x^2-8x-7\right)+\left(-7x^3-9x\right)$(3`x`3−9`x`2−8`x`−7)+(−7`x`3−9`x`).

If a picture frame has a length of $8x^2-9x+3$8`x`2−9`x`+3 and a width of $6x^3+9x^2$6`x`3+9`x`2, form a fully simplified expression for the perimeter of the rectangular picture frame.

If $P\left(x\right)=3x^2+7x-6$`P`(`x`)=3`x`2+7`x`−6 and $Q\left(x\right)=6x-7$`Q`(`x`)=6`x`−7, form a simplified expression for $P\left(x\right)-Q\left(x\right)$`P`(`x`)−`Q`(`x`).