7A - Introduction to polynomials
7A - Content video: Introduction to polynomials
This video covers the theory and an example relating to polynomial terms and functions.
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■ A polynomial term has the form \(ax^n\), where \(a\in \mathbb{R}\) and \(n\in {0,1,2,3,4,...}\)
Examples of polynomial functions include:
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The following are not polynomial functions:
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Describing polynomials:
- The leading term contains the variable, \(x\), raised to the highest power.
- A monic polynomial has a leading term with a coefficient of \(1\).
- The degree of the polynomial is determined by the highest power present.
Arithmetic of polynomials:
Polynomials can be added, subtracted, multiplied and even divided.
Polynomials can be added, subtracted, multiplied and even divided.
- When two polynomials, \(P\) and \(Q\), are added or subtracted the degree of the resulting polynomial is less than or equal to the degree of the starting polynomial with the highest order.
- When two polynomials, \(P\) and \(Q\), are multiplied the degree of the resulting polynomial is equal to the sum of the degrees of the starting polynomial with the highest order.
7A - Example 1: Arithmetic of polynomials
Consider the polynomials \(P(x)=3x^3+4x-3\) and \(Q(x)=x^3-7x\), find:
(a) \(P(x)+Q(x)\) (b) \(P(x)-Q(x)\) (c) \(P(x)\times Q(x)\) |
7A - Example 1: Video solution
7A - Example 1: Practice
Question 1: ABC Question 2: ABC 7A - Example 1: Solutions
Question 1: ABC Question 2: ABC |
Equating coefficients:
For two polynomials to be equivalent, the corresponding coefficients must be equal.
For two polynomials to be equivalent, the corresponding coefficients must be equal.
7A - Example 2: Equating coefficients of polynomial functions
If \(y=ax^3+bx^2+cx+d\) is equal to \(P(x)=2x^3+4x-3\), state the value of \(a\), \(b\), \(c\) and \(d\).
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7A - Example 2: Video solution
7A - Example 2: Practice
Question 1: ABC Question 2: ABC 7A - Example 2: Solutions
Question 1: ABC Question 2: ABC |
7A - Example 3: Equating coefficients of polynomial functions
The polynomial \(Q(x)=2x^3-6x^2+6x+2\) can be written in the form \(A(x-h)^3+k\), where \(A\), \(h\) and \(k\) are real numbers. Find the values of \(A\), \(h\) and \(k\).
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7A - Example 3: Video solution
7A - Example 3: Practice
Question 1: ABC Question 2: ABC 7A - Example 3: Solutions
Question 1: ABC Question 2: ABC |