Revision:OCR Core 1 - Polynomials

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Revision:OCR Core 1 - Polynomials

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > OCR Core 1 - Polynomials

1 9.Polynomials
- 1.1 Polynomials
- 1.2 Addition, subtraction and multiplication of polynomials
2 Also See
3 Comments

9.Polynomials

Polynomials

A polynomial is an expression which consists of a sum (although one must realise that this can mean that there are "subtracted" terms) of multiples of powers of a variable, and constant terms.

A quadratic expression is a polynomial that is common, and hence has a name given to it.

One piece of information that one must know about polynomials is the definition of the term degree. One says that a polynomial has a degree of 2 if the greatest power of a variable within it is 2. To generalise, an "nth" degree polynomial, is a polynomial whose highest variable power is "n".

Each of the different powers in the sum, and the constant value are called the terms; for instance:

$ax^{3} + bx^{2} + cx + d$

This is an example of a polynomial whose degree is "3", and has four terms (namely, $ax^{3}$ , $bx^{2}$ , , and ).

The values of a, b, and c are the coefficients, and the leading coefficient is a (this is the coefficient which is of the highest power of the variable). The constant term is d.

Now one can learn the names of some of the named polynomials:

A polynomial of degree 0 is called a constant (this is because it can only consist of constant terms).

A polynomial of degree 1 is called a linear polynomial as it has only a term of "x", and possibly a constant.

A polynomial of degree 2 is called a quadratic polynomial as it has an $x^{2}$ term.

A polynomial of degree 3 is called a cubic polynomial as it has an $x^{3}$ term.

A polynomial of degree 4 is called a quartic polynomial as it has an $x^{4}$ term.

A polynomial of degree 5 is called a quintic polynomial as it has an $x^{5}$ term.

Sometimes it is requested that one gives a polynomial in ascending order, or descending order. Ascending order is simply when one displays the terms in order of their power, hence one would display the polynomial that was given earlier in the notes in this section, as:

$d + cx + bx^{2} + ax^{3}$ .

Descending order is merely the opposite, and hence the resultant expression is:

$ax^{3} + bx^{2} + cx + d$ .

Fractional, or negative powers of a variable cannot be in a polynomial.

Addition, subtraction and multiplication of polynomials

The addition and subtraction of polynomials is a simple task. One must merely add, and subtract the like terms, hence one should not (and cannot) add $x^{2}$ terms to $x^{3}$ terms, as they are not like terms (they are unlike terms).

Example

1. $p(x) = x^{3} + 2x^{2} + 4x + 8$

$q(x) = 2x^{3} + 6x^{2} + 18x + 54$

Calculate:

p(x) + q(x) ,

and

q(x) - p(x) .

This is a simple process, simply collect, and use the corresponding operation on the like terms.

$p(x) + q(x) = \left( x^{3} + 2x^{2} + 4x + 8 \right) + \left( 2x^{3} + 6x^{2} + 18x + 54 \right) = 3x^{3} + 8x^{2} + 22x + 62$

$q(x) - p(x) = \left( 2x^{3} + 6x^{2} + 18x + 54 \right) - \left( x^{3} + 2x^{2} + 4x + 8 \right) = x^{3} + 4x^{2} + 14x + 46$

One thing that one should be vigilant of is that often people make the mistake of subtracting all the terms in the polynomial (when they are asked to conduct a subtraction), regardless of the sign. If there is a subtraction of a negative value, it becomes an addition.

Multiplication of polynomials is slightly more difficult, but if one remembers that one must multiply all of the terms, one should not have trouble.

The most simple type of multiplication of a polynomial is multiplication by a constant, evidently giving:

$a \left( ax^{3} + bx^{2} + cx + d \right) = aex^{3} + bex{2} + cex + de$

Another easy multiplication is that of a power of a variable, for example:

$x^{2} \left( ax^{3} + bx^{2} + cx + d \right) = ax^{5} + bx^{4} + cx^{3} + dx^{2}$

Now one must consider that if one is to multiply one polynomial by another, one should multiply all terms in one of the polynomials by all terms in the other, for example:

$(ax + b)(cx + d) = acx^{2} + adx + bcx + bd$

It is a good idea to work in a logical manner, such that one does not miss out terms.

It may be asked that you calculate the coefficient of a specific term of a multiplication. One could do the full multiplication, however, it can be a lot easier to simply spot the pairs (in the context of two polynomials being multiplied) that will give the desired term, and then calculate what the sum of the relevant coefficient products is.

Example

1. Calculate the coefficient of $x^{2}$ in the following multiplication:

$(x^{2} + 2x + 3)(4x^{2} + 5x + 6)$

One should note that one can obtain a term of $x^{2}$ from a constant multiplied by $x^{2}$ , or a term of multiplied by a term of .

Hence one should calculate the relevant coefficients (in this example the term of the variable is left in to demonstrate, and aid the following of the example, it is generally redundant in normal work, and hence one might opt to disregard it):

$x^{2} \times 6 = 6x^{2}$

$4x^{2} \times 3 = 12x^{2}$

$2x \times 5x = 10x^{2}$

Hence the coefficient of $x^{2}$ will be 6 + 12 + 10 = 28.

One rule which is useful to know is that if one is to multiply two polynomials, the degree of the resultant polynomial is the sum of the degrees of the two polynomials who are being multiplied (this is easily explained through the "multiplication rule" for indices).