What is the factor theorem ?

The factor theorem is used to find the linear factors of polynomial equations. It states that for any polynomial function P ( x ): ( x - k ) is a factor of P ( x ) if P ( k ) = 0 P ( k ) = 0 if ( x - k ) is a factor of P ( x )

What is the remainder theorem ?

The remainder theorem is used to find the remainder when we divide a polynomial function by a linear function . It states that when any polynomial P ( x ) is divided by any linear function ( x - k ) the value of the remainder R is given by P ( k ) = R.

What is polynomial division ?

Polynomial division is the process of dividing two polynomials . Note that this is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator .

True or False? In a polynomial division, the quotient is the sum of the terms that the divisor was multiplied by.

True. In a polynomial division, the quotient is the sum of the terms that the divisor was multiplied by.

What value in polynomial division is the remainder ?

In polynomial division, subtractions are carried out until all terms in the original polynomial being divided are cancelled out. Any left over polynomial function is the remainder .

True or False? When dividing a polynomial by a linear function the remainder will always be a constant .

True. When dividing a polynomial by a linear function the remainder will always be a constant.

True or False? Every real polynomial of odd degree has at least one real zero .

True. Every real polynomial of odd degree has at least one real zero .

What three things do you need to know to sketch the graph of a polynomial without a GDC?

The three things you need to know to sketch the graph of a polynomial without a GDC, are: the y -intercept, the roots, and the shape.

How can you find the y -intercept of a polynomial?

You can find the y -intercept of a polynomial by substituting x = 0 to get y = a 0 .

How can you find the roots of a polynomial?

You can find the roots of a polynomial by factorising or solving y = 0 .

True or False? The shape of a graph is determined only by the sign of the leading coefficient ( a n ).

False. The shape of a graph is determined by the sign of the leading coefficient ( a n ) and by the degree (n) of the equation.

What is meant by the multiplicity of a root in the context of polynomials?

The multiplicity of a root is the number of times it is repeated when the polynomial is factorised.

What is the ' Fundamental Theorem of Algebra '?

The Fundamental Theorem of Algebra states that every real polynomial with degree n can be factorised into n complex linear factors . Some of these may be repeated factors.

How can you find another root if you know a complex root of a real polynomial ?

If you know a complex root of a real polynomial then its complex conjugate will always be another root .

How can you find unknowns using the sum and product of roots ?

To find unknowns using the sum and product of roots, form two equations using the formulae and solve .

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Polynomial Functions (DP IB Analysis & Approaches (AA))

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FrontFactor & Remainder Theorem
What is the factor theorem?
BackFactor & Remainder Theorem
The factor theorem is used to find the linear factors of polynomial equations.
It states that for any polynomial function P(x):
(x - k) is a factor of P(x) if P(k) = 0
P(k) = 0 if (x - k) is a factor of P(x)

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Cards in this collection (32)

What is the factor theorem?
The factor theorem is used to find the linear factors of polynomial equations.
It states that for any polynomial function P(x):
- (x - k) is a factor of P(x) if P(k) = 0
- P(k) = 0 if (x - k) is a factor of P(x)
True or False?
For any polynomial function $P (x)$ , $(x - k)$ is a factor of $P (x)$ if $P (k) = 0$ .
True.
For any polynomial function $P (x)$ , $(x - k)$ is a factor of $P (x)$ if $P (k) = 0$ .
Given a polynomial function $f (x)$ , how can the factor theorem be used to determine if $(x - a)$ is a linear factor?
For a polynomial function $f (x)$ , the factor theorem states that if $f (a) = 0$ then $(x - a)$ must be a linear factor.
I.e. if you substitute $x = a$ into the original function, $f (x)$ , and the result is 0, then $(x - a)$ is a factor of $f (x)$ .
What value must be substituted into a polynomial function $f (x)$ to determine if $(a x + b)$ is a linear factor of $f (x)$ ?
The value that must be substituted into the polynomial function $f (x)$ to determine if $(a x + b)$ is a linear factor, is $(- \frac{b}{a})$ .
What is the remainder theorem?
The remainder theorem is used to find the remainder when we divide a polynomial function by a linear function.
It states that when any polynomial P(x) is divided by any linear function (x - k) the value of the remainder R is given by P(k) = R.
When any polynomial $P (x)$ is divided by any linear function $(x - k)$ , what is the value of the remainder $R$ ?
When any polynomial $P (x)$ is divided by any linear function $(x - k)$ , the value of the remainder $R$ is $P (k)$ .
I.e. if you substitute $x = k$ into the original function, the result will be the remainder.
If a polynomial function $f (x)$ is divided by a linear function $(a x + b)$ , what value must be substituted to determine the remainder?
The value that must be substituted into the polynomial function $f (x)$ to determine the remainder when $f (x)$ is divided by a linear function of the form $(a x + b)$ , is $(- \frac{b}{a})$ .
What is polynomial division?
Polynomial division is the process of dividing two polynomials.
Note that this is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator.
When performing a polynomial division, e.g. $\frac{2 x^{5} - x^{4} + 3 x^{2} + 8}{x + 4}$ , how do you determine the value to subtract from the numerator in order to cancel out the first term ?
When performing a polynomial division, $\frac{2 x^{5} - x^{4} + 3 x^{2} + 8}{x + 4}$ , to determine the value to subtract from the numerator in order to cancel out the first term, you have to:
- divide the leading term of the polynomial by the leading term of the divisor, e.g. $\frac{2 x^{5}}{x} = 2 x^{4}$ ,
- and multiply this by the divisor, e.g. $2 x^{4} (x + 4) = 2 x^{5} + 8 x^{4}$ .
This value can now be subtracted from the polynomial to cancel out the first term.
How can a polynomial function with some missing power terms, e.g. $x^{4} - 2 x^{3} + x - 7$ , be adapted to make polynomial division easier?
A polynomial function with some missing power terms, can be rewritten with 0's to be used as placeholders, to make polynomial division easier.
E.g. $x^{4} - 2 x^{3} + x - 7$ can be re-written as $x^{4} - 2 x^{3} + 0 x^{2} + x - 7$ .
True or False?
In a polynomial division, the quotient is the sum of the terms that the divisor was multiplied by.
True.
In a polynomial division, the quotient is the sum of the terms that the divisor was multiplied by.
What value in polynomial division is the remainder?
In polynomial division, subtractions are carried out until all terms in the original polynomial being divided are cancelled out.
Any left over polynomial function is the remainder.
What is an alternative method to polynomial division you can use to find quotients and remainders?
An alternative method to polynomial division you can use to find quotients and remainders is to compare coefficients.
Given a polynomial, $P (x)$ , and a divisor, $D (x)$ , you can write $P (x) = Q (x) D (x) + R (x)$ , where $Q (x)$ is the quotient and $R (x)$ is the remainder, then
- expand the right-hand side,
- equate the coefficients,
- and solve to find the unknown values.
True or False?
When dividing a polynomial by a linear function the remainder will always be a constant.
True.
When dividing a polynomial by a linear function the remainder will always be a constant.
True or False?
When dividing a polynomial by a quadratic function the remainder will always be of the form $a x + b$ .
False.
When dividing a polynomial by a quadratic function the remainder will not always be of the form $a x + b$ .
The remainder will either be linear $(a x + b)$ or a constant.
If a real polynomial $P (x)$ has degree $n$ , how many zeros will it have?
If a real polynomial $P (x)$ has degree $n$ , it will have $n$ zeros, which can be written in the form $a + b i$ , where $a, b \in ℝ$ .
True or False?
Every real polynomial of odd degree has at least one real zero.
True.
Every real polynomial of odd degree has at least one real zero.
What three things do you need to know to sketch the graph of a polynomial without a GDC?
The three things you need to know to sketch the graph of a polynomial without a GDC, are:
- the y-intercept,
- the roots,
- and the shape.
How can you find the y-intercept of a polynomial?
You can find the y-intercept of a polynomial by substituting x = 0 to get y = a₀.
How can you find the roots of a polynomial?
You can find the roots of a polynomial by factorising or solving y = 0.
True or False?
The shape of a graph is determined only by the sign of the leading coefficient (a_n).
False.
The shape of a graph is determined by the sign of the leading coefficient (a_n) and by the degree (n) of the equation.
What is the maximum number of stationary points for a polynomial of degree $n$ ?
The maximum number of stationary points for a polynomial of degree $n$ is $(n - 1)$ .
What is meant by the multiplicity of a root in the context of polynomials?
The multiplicity of a root is the number of times it is repeated when the polynomial is factorised.
How does a root with multiplicity 2 affect the graph of a polynomial?
If $x = k$ has multiplicity 2, then the graph has a turning point at $(k, 0)$ so touches the x-axis (rather than intersecting it).
True or False?
The graph shown below has a root with multiplicity 3.
True.
The graph shown has a root with multiplicity 3 as it shows a curve crossing the x-axis at a point of inflection.
What is the 'Fundamental Theorem of Algebra'?
The Fundamental Theorem of Algebra states that every real polynomial with degree n can be factorised into n complex linear factors.
Some of these may be repeated factors.
True or False?
If $a + b i$ is a zero of a real polynomial, then $a - b i$ is also a zero.
True.
If $a + b i$ , $(b \neq 0)$ , is a zero of a real polynomial, then $a - b i$ is also a zero.
What is an irreducible quadratic?
An irreducible quadratic is a quadratic factor that does not have real roots. Its discriminant will be negative: $b^{2} - 4 a c < 0$ .
State the formula for the sum of roots of a polynomial.
The formula for the sum of roots of a polynomial is $- \frac{a_{n - 1}}{a_{n}}$ ,
Where:
- $a_{n}$ is the coefficient of the leading term.
- $a_{n - 1}$ is the coefficient of the $x^{n - 1}$ term.
This is given in your exam formula booklet.
State the formula for the product of roots of a polynomial.
The formula for the product of roots of a polynomial is $\frac{{(- 1)}^{n} a_{0}}{a_{n}}$ ,
Where:
- $a_{n}$ is the coefficient of the leading term.
- $a_{0}$ is the constant term.
This is given in your exam formula booklet.
How can you find another root if you know a complex root of a real polynomial?
If you know a complex root of a real polynomial then its complex conjugate will always be another root.
How can you find unknowns using the sum and product of roots?
To find unknowns using the sum and product of roots, form two equations using the formulae and solve.

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