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Algebra - Classification of polynomials

Grade 9IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, 4x3−2x+74x^3 - 2x + 7 is a polynomial, but x−2x^{-2} or x\sqrt{x} are not.

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The degree of a polynomial is the highest power of the variable in the expression. For a polynomial with multiple variables, the degree of a term is the sum of the exponents of the variables in that term.

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Classification by Number of Terms: A Monomial has one term (e.g., 5x25x^2), a Binomial has two terms (e.g., 2x+32x + 3), and a Trinomial has three terms (e.g., x2−4x+4x^2 - 4x + 4).

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Classification by Degree: A polynomial of degree 0 is a Constant (kk), degree 1 is Linear (ax+bax + b), degree 2 is Quadratic (ax2+bx+cax^2 + bx + c), degree 3 is Cubic (ax3+bx2+cx+dax^3 + bx^2 + cx + d), and degree 4 is Quartic.

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The Leading Coefficient is the coefficient of the term with the highest degree, and the Constant Term is the term with no variable (degree 0).

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A polynomial is in Standard Form when its terms are written in descending order of their exponents (from highest degree to lowest degree).

📐Formulae

P(x)=anxn+an−1xn−1+⋯+a1x+a0P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0

Linear: f(x)=ax+b,a≠0\text{Linear: } f(x) = ax + b, \quad a \neq 0

Quadratic: f(x)=ax2+bx+c,a≠0\text{Quadratic: } f(x) = ax^2 + bx + c, \quad a \neq 0

Cubic: f(x)=ax3+bx2+cx+d,a≠0\text{Cubic: } f(x) = ax^3 + bx^2 + cx + d, \quad a \neq 0

💡Examples

Problem 1:

Classify the polynomial 7x2−5x3+27x^2 - 5x^3 + 2 by its degree and the number of terms.

Solution:

Degree: 33 (Cubic); Number of terms: 33 (Trinomial).

Explanation:

First, rewrite the polynomial in standard form: −5x3+7x2+2-5x^3 + 7x^2 + 2. The highest exponent is 33, making it a cubic polynomial. Since there are three distinct terms separated by plus/minus signs, it is a trinomial.

Problem 2:

Identify the leading coefficient and the degree of the polynomial P(y)=12−4y+9y5P(y) = 12 - 4y + 9y^5.

Solution:

Leading Coefficient: 99; Degree: 55.

Explanation:

The term with the highest power of yy is 9y59y^5. The exponent is 55, which represents the degree. The coefficient of this term is 99, which is the leading coefficient.

Problem 3:

Determine if the expression 3x2+5x−13x^2 + \frac{5}{x} - 1 is a polynomial.

Solution:

No, it is not a polynomial.

Explanation:

In a polynomial, all variables must have non-negative integer exponents. The term 5x\frac{5}{x} can be written as 5x−15x^{-1}. Since the exponent −1-1 is a negative integer, the expression does not meet the definition of a polynomial.