Degree of a Polynomial

This is the zero polynomial and its degree is not defined. In exams, identify it separately from non-zero constants.

The highest exponent of \(x\) is 3, so the polynomial has degree 3 and is cubic. The term \(-4x\) has exponent 1, so it does not determine the classification. Exam tip: identify the highest exponent first.

Combining like terms gives \(8x^6-3x^6=5x^6\), so the polynomial becomes \(5x^6+2x^3-7\). The greatest exponent of the variable is 6; therefore, its degree is 6. Option 5 confuses the coefficient in \(5x^6\) with the degree. The degree is the highest power of the variable, not its coefficient. Exam tip: simplify like terms first, then identify the highest exponent among the non-zero terms.

The degree of a polynomial is the greatest exponent of the variable in a term whose coefficient is non-zero. Since \(r\neq0\), \(rx^8\) is a non-zero term. The greatest power of \(x\) in the polynomial is \(8\), so its degree is \(8\). Option 3 represents the exponent of \(x^3\), but it is not the highest exponent. Exam tip: Before finding degree when a parameter is present, check whether its coefficient is zero or non-zero.

When \(r=0\), \(rx^8=0\cdot x^8=0\). The expression therefore becomes \(-5x^3+2\). The highest power of \(x\) with a non-zero coefficient is 3, so the degree is 3. Option 8 is incorrect because the \(x^8\) term vanishes when its coefficient is zero. Exam tip: remove all terms with zero coefficients before finding the degree.

The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. Here, the exponents of the terms are 4, 2, and 0, and the greatest is 4. Therefore, the correct answer is 4. The number 12 is a constant term, so its exponent is 0; it does not determine the degree. Exam tip: To find degree, look for the highest power of the variable, not the numerical coefficient.

The governing concept is that the degree of a polynomial is determined with respect to the stated variable by the greatest exponent of that variable having a non-zero coefficient. Here the stated variable is x, but the expression 7y^3 - 2y + 5 contains no x-term at all. Therefore, when regarded as a polynomial in x, every term is independent of x and the whole expression acts as the non-zero constant 7y^3 - 2y + 5. A non-zero constant polynomial has degree 0, so option C is correct. Option A would be the degree in y, and option B would incorrectly treat the first power of y as decisive. Option D is not suitable because the expression is still a well-defined constant polynomial in x.

The degree of a polynomial is the greatest exponent of the variable whose coefficient is non-zero. In \(7x^3-2x+1\), the highest power of \(x\) is 3, so its degree is 3. \(4x^2-7x+9\) has degree 2, while \(x^4+x^2-5\) has degree 4. \(12\) is a non-zero constant polynomial and has degree 0. Exam tip: first identify the term with the greatest power after simplifying the polynomial.

On expanding, multiply 3x^3 by both terms: 3x^3(2x^2-5)=6x^5-15x^3. The highest power of x is 5, so the degree of the polynomial is 5. Here, 3 is the power in the second term, while 6 is a coefficient, not a degree. Exam tip: after simplifying, the greatest exponent of the variable with a non-zero coefficient gives the degree.

First expand: \(x^3(x^2+2)=x^5+2x^3\). Thus, the expression becomes \(x^5+2x^3-x^5+7x^2=2x^3+7x^2\). The \(x^5\) terms cancel, and the highest power of \(x\) in the remaining polynomial is 3. Therefore, its degree is 3. Option 5 is incorrect because the \(x^5\) term does not remain after simplification. Exam tip: Always simplify and combine like terms before finding the degree of a polynomial.

\(-5\) is a non-zero constant polynomial. Since it contains no variable with a positive power, its degree is \(0\). The degree of \(x\) is \(1\), while that of \(x^2+1\) is \(2\). \(0\) is the zero polynomial, whose degree is not defined. Exam tip: degree \(0\) applies only to non-zero constant polynomials.

In a multivariable polynomial, a term’s degree is the sum of the exponents of its variables. For \(6x^2y^3\), it is \(2+3=5\), the greatest term degree. Looking only at the exponent of \(y\) is incorrect. Exam tip: add exponents in each term.

A polynomial whose highest power of the variable is 1 has degree 1 and is called a linear polynomial. For example, \(3x-5\) is a linear polynomial. A constant polynomial has degree 0, while quadratic and cubic polynomials have degrees 2 and 3 respectively. Exam tip: identify the type of a polynomial by its highest power of the variable.

Combining like terms gives 2x^5-5x^5=-3x^5. Thus, f(x)=-3x^5+3x^2-8. The highest power of x is 5, so the degree of the polynomial is 5. It is not 2 because the x^5 term does not cancel completely. Exam tip: simplify by combining like terms first, then identify the highest remaining exponent.

The total degree of a polynomial is the greatest sum of the exponents of variables in any one term. For \(5x^2y^3\), the sum is \(2+3=5\); for \(4xy\), it is \(1+1=2\); and the constant \(-6\) has degree 0. Therefore, the polynomial has total degree 5. Option 2 is only the degree of \(4xy\), not of the whole polynomial. Exam tip: Add the exponents within each term and select the greatest sum.

For a multivariable polynomial, the degree of a term is the sum of the exponents of all variables in that term. In 7x³y², the term degree is 3 + 2 = 5. In −2xy⁴, it is 1 + 4 = 5, while the constant 10 has degree 0. The polynomial’s total degree is the greatest term degree, so it is 5. The coefficient 7 is not the degree.

To find the degree with respect to (x), treat (y) as a constant coefficient. The powers of (x) in the expression are 3, 1, and 0. The greatest power is 3, so the degree is 3. Although (y) has power 5, the degree is not 5 because the question asks specifically with respect to (x). Exam tip: In a multivariable expression, first identify the variable with respect to which the degree is required.

When the polynomial is considered with respect to \(y\), \(x\) is treated as part of the coefficient. The powers of \(y\) in the terms are \(2\), \(5\), and \(0\). The greatest of these is \(5\), so the degree with respect to \(y\) is \(5\). Choosing \(3\) would be incorrect because it is the power of \(x\), not of \(y\). Exam tip: For degree with respect to one variable, look only at the highest power of that variable.

Terms with zero coefficients disappear and (-12) remains. (-12) is a non-zero constant polynomial of degree (0).

For degree (2), the (x^3) term must vanish and the (x^2) term must remain. At (t=-4), both conditions hold.

When a parameter appears in the coefficients, its given value must be substituted before finding the degree. The degree is then the greatest exponent of \(x\) attached to a non-zero coefficient. A term with a high exponent does not count if its coefficient becomes zero after substitution. This is the key point in this expression.

For \(u=2\), the coefficient of \(x^6\) is \(u-2=2-2=0\), so the \(x^6\) term disappears. The coefficient of \(x^4\) is \(u^2+1=2^2+1=5\), which is non-zero. The remaining term \(x\) has degree \(1\). Thus the highest remaining exponent is \(4\), so option B is correct.

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. In \(7x^4-x+2\), the highest power of \(x\) is 4, so it is a polynomial of degree four. The closest distractor, \(2x^5+x-1\), has degree 5, not 4. Exam tip: identify the term containing the highest power of the variable.

Here, x, (x+4), and (x-5) are three linear factors, each of degree 1. When polynomials are multiplied, their degrees add: 1+1+1=3. Therefore, the degree of the polynomial is 3. Option 2 would apply to a product of only two linear factors. Exam tip: For a polynomial given in factorised form, add the degrees of its factors.

On combining like terms, \(5x^7-5x^7=0\). Thus, \(g(x)=x^5-x^2\). The highest exponent having a non-zero coefficient is 5, so the degree of the polynomial is 5. Choosing 7 would be incorrect because the \(x^7\) terms cancel each other. Exam tip: Always simplify like terms before finding the degree.

The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. In the given polynomial, the powers of x are 3, 5, 7, and 9; the greatest is 9. Hence, its degree is 9. Although 7 is a close distractor, the term x^9 is present, so the degree cannot be 7. Exam tip: identify the highest power of the variable among all the terms.