- Use the Remainder Theorem to evaluate polynomials and apply the Factor Theorem to find roots.
- Apply the Rational Zero Theorem and Descartes’ Rule of Signs to identify possible rational zeros and estimate the number of positive and negative real zeros.
- Find zeros of polynomial functions and use the Linear Factorization Theorem to construct polynomials.
- Solve real-life problems using polynomial equations.
Remainder Theorem
The Main Idea
The Remainder Theorem is a quick way to evaluate polynomials. If you divide a polynomial [latex]f(x)[/latex]by [latex]x - k[/latex], the remainder is simply the value of [latex]f(k)[/latex]. It’s like a shortcut in a maze, allowing you to reach the end without going through all the paths.
For example, to evaluate [latex]f(x) = 6x^4 - x^3 - 15x^2 + 2x - 7[/latex] at [latex]x = 2[/latex], you can use synthetic division or directly substitute [latex]x[/latex] with [latex]2[/latex] to find the remainder.
Watch the following video for more on the remainder theorem.
You can view the transcript for “Remainder Theorem and Synthetic Division of Polynomials” here (opens in new window).
You can view the transcript for “Ex 2: Find the Zeros of a Polynomial Function – Real Rational Zeros” here (opens in new window).
Rational Zero Theorem
The Main Idea
When we’re navigating the complex sea of polynomial functions, the Rational Zero Theorem is like our compass. It helps us pinpoint potential rational zeros using a simple yet powerful strategy: looking at the factors of the constant term and the leading coefficient.
For instance, consider a polynomial function that has zeros at [latex]\frac{2}{5}[/latex] and [latex]\frac{3}{4}[/latex]. These aren’t just numbers; they’re clues. By setting up equations with these zeros and constructing a quadratic function, we can see a pattern emerge. The numerators of these zeros ([latex]2[/latex] and [latex]3[/latex]) are factors of the constant term, while the denominators ([latex]5[/latex] and [latex]4[/latex]) are factors of the leading coefficient.
Quick Tips: Applying the Rational Zero Theorem
- List Out Factors: Start by listing all factors of the constant term and the leading coefficient of your polynomial.
- Form Ratios: Create all possible ratios [latex]\frac{p}{q}[/latex] where [latex]p[/latex] is a factor of the constant term and [latex]q[/latex] is a factor of the leading coefficient.
- Test Your Candidates: Evaluate each potential zero by plugging it into the polynomial. If the result is zero, you’ve found a true zero!
Watch the following video see more on the rational zero theorem.
You can view the transcript for “Finding All Zeros of a Polynomial Function Using The Rational Zero Theorem” here (opens in new window).
You can view the transcript for “Ex 1: Find the Zeros of a Polynomial Function – Integer Zeros” here (opens in new window).
Factor Theorem
The Main Idea
The Factor Theorem is a bridge between zeros and factors of a polynomial. It states that a number [latex]k[/latex] is a zero of [latex]f(x)[/latex] if and only if [latex](x - k)[/latex] is a factor of [latex]f(x)[/latex]. This theorem is like a two-way street; knowing a zero lets you find a factor, and knowing a factor lets you find a zero.
For instance, to show that [latex](x + 2)[/latex] is a factor of [latex]x^3 - 6x^2 - x + 30[/latex], you can use synthetic division. If the remainder is zero, then [latex](x + 2)[/latex] is indeed a factor, and you can further factorize the quotient to find the remaining zeros of the polynomial.
You can view the transcript for “(New Version Available) Polynomial Function – Complex Factorization Theorem” here (opens in new window).
The Fundamental Theorem of Algebra
The Main Idea
- Every non-constant polynomial function has at least one complex zero.
- For a polynomial of degree [latex]n > 0[/latex], it can be written as: [latex]f(x) = a(x-c_1)(x-c_2)...(x-c_n)[/latex] where [latex]a[/latex] is a non-zero real number and [latex]c_1, c_2, ..., c_n[/latex] are complex numbers.
- A polynomial of degree [latex]n[/latex] has exactly [latex]n[/latex] roots, counting multiplicities.
- Complex zeros can be real or imaginary.
- The Fundamental Theorem of Algebra is crucial for solving polynomial equations.
You can view the transcript for “Fundamental theorem of algebra | Polynomial and rational functions | Algebra II | Khan Academy” here (opens in new window).
Linear Factorization Theorem and Complex Conjugate Theorem
The Main Idea
- Linear Factorization Theorem:
- A polynomial of degree n has exactly n linear factors.
- Each factor is of the form [latex](x - c)[/latex], where c is a complex number.
- Complex Conjugate Theorem:
- For polynomials with real coefficients, complex zeros always occur in conjugate pairs.
- If [latex]a + bi[/latex] is a zero, then [latex]a - bi[/latex] is also a zero.
- Factoring polynomials:
- Real zeros produce factors like [latex](x - r)[/latex] where r is real.
- Complex zeros produce pairs of factors: [latex](x - (a+bi))(x - (a-bi))[/latex]
- Reconstructing polynomials:
- Given the zeros and a point on the graph, you can reconstruct the entire polynomial.
You can view the transcript for “Find a Polynomial with Real Coefficients that has the Given Zeros” here (opens in new window).
Descartes’ Rule of Signs
The Main Idea
- Descartes’ Rule of Signs provides a method to determine the possible numbers of positive and negative real zeros in a polynomial function.
- For positive real zeros:
- Count the number of sign changes in [latex]f(x)[/latex] when written in descending order.
- The number of positive real zeros is either equal to this count or less than it by an even integer.
- For negative real zeros:
- Replace [latex]x[/latex] with [latex]-x[/latex] in [latex]f(x)[/latex] to get [latex]f(-x)[/latex].
- Count the number of sign changes in [latex]f(-x)[/latex].
- The number of negative real zeros is either equal to this count or less than it by an even integer.
- The rule doesn’t give the exact number of real zeros, but provides upper bounds and parity information.
You can view the transcript for “How to use Descartes rule of signs to determine the number of positive and negative zeros” here (opens in new window).