Define algebraic division , and say what it is used for.

Algebraic division, also called polynomial division , splits a polynomial into a pair of factors, with or without a remainder term. It is used to factorise polynomials and to simplify algebraic fractions.

True or False? A remainder of zero in an algebraic division means the divisor is a factor of the polynomial.

True. A zero remainder means the polynomial splits exactly into the divisor multiplied by the quotient, with nothing left over. That is precisely what it means for the divisor to be a factor .

IGCSEMathsEdexcelMaths BFlashcardsAlgebraThe Factor Theorem & Algebraic Division

The Factor Theorem & Algebraic Division (Edexcel IGCSE Maths B): Flashcards

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FrontThe Factor Theorem
State the factor theorem.

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Cards in this collection (12)

State the factor theorem.
For a function $f (x)$ , if $f (a) = 0$ then $(x - a)$ is a factor of $f (x)$ .
The theorem works both ways, so if $(x - a)$ is a factor then $f (a) = 0$ .
$(x + 4)$ is a factor of $f (x)$ . Fill in the missing value.
$f (______) = 0$
The completed statement is:
$f (- 4) = 0$
A factor written $(x - a)$ gives $f (a) = 0$ , and $(x + 4)$ is $(x - (- 4))$ , so here $a$ is $- 4$ rather than $4$ .
Testing whether $(x + 1)$ is a factor of $g (x)$ gives $g (- 1) = 7$ . What does that tell you?
It tells you that $(x + 1)$ is not a factor of $g (x)$ .
The factor theorem needs the substitution to give exactly zero, so any other value rules the factor out.
True or False?
To test whether $(2 x - 3)$ is a factor of $h (x)$ , you work out $h (3)$ .
False.
When the linear factor has a coefficient of $x$ , take that coefficient out first: $(2 x - 3) = 2 (x - \frac{3}{2})$ .
The factor to test is therefore $(x - \frac{3}{2})$ , so it is $h (\frac{3}{2})$ that must equal zero.
If $(x - a)$ is a factor of $f (x)$ , what does dividing $f (x)$ by it give?
It gives the quotient $Q (x)$ , so that $f (x) = (x - a) \times Q (x)$ .
The quotient is itself a factor of $f (x)$ , so factorising it further gives any remaining factors.
Where should you look first for integer solutions of a polynomial equation?
Among the factors of the constant term, because an integer solution of a polynomial equation with whole-number coefficients must divide that constant.
For $x^{3} - 2 x^{2} - x + 2$ the constant is $2$ , so the only integer solutions possible are $\pm 1$ and $\pm 2$ .
Define algebraic division, and say what it is used for.
Algebraic division, also called polynomial division, splits a polynomial into a pair of factors, with or without a remainder term.
It is used to factorise polynomials and to simplify algebraic fractions.
Fill in the names of the two parts of an algebraic division.
$A division works out the______divided by the______to give a quotient$
The completed sentence is:
A division works out the dividend divided by the divisor to give a quotient.
On this course the dividend is a cubic and the divisor is a linear term, usually of the form $(x \pm a)$ or $(a x \pm b)$ .
Define a polynomial.
A polynomial is an algebraic expression made up of a finite number of terms whose powers are all non-negative whole numbers.
So $2 x^{2} y - 4 y + 6$ is a polynomial, and so is a plain number such as $8$ , while $\sqrt{x}$ and $\frac{3}{x}$ are not.
In algebraic long division, what happens after each term of the answer is written?
That term is multiplied by the divisor, and the result is subtracted from what is left.
The answer is built up term by term, working downwards in powers of $x$ and starting from the highest power.
When does an algebraic long division stop, and what is left at that point?
It stops once what is left is just a number, with no $x$ terms in it.
That number is the remainder.
True or False?
A remainder of zero in an algebraic division means the divisor is a factor of the polynomial.
True.
A zero remainder means the polynomial splits exactly into the divisor multiplied by the quotient, with nothing left over.
That is precisely what it means for the divisor to be a factor.

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