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 degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In \(6x^5-2x+7\), the greatest exponent of \(x\) is 5, so its degree is 5. Option A has degree 4, while option C has degree 6. Option D is a non-zero constant polynomial, so its degree is 0. Exam tip: identify the highest power of the variable to find the degree.

Combining like terms gives \(9x^7-4x^7=5x^7\). Thus, the simplified polynomial is \(5x^7+5x^3-2\). The highest power of \(x\) is \(7\), so its degree is 7. Option 5 is the coefficient of \(x^7\), not the degree. Exam tip: simplify like terms first, then identify the greatest exponent with a non-zero coefficient.

The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. Since \(r\neq0\), the term \(rx^9\) is non-zero. Hence, the highest exponent of \(x\) is \(9\), so the degree of the polynomial is \(9\). The number \(4\) is the degree of the term \(2x^4\), not of the entire polynomial. Exam tip: always check whether the coefficient of the highest-power term is zero.

On substituting \(r=0\), \(rx^9=0\cdot x^9=0\). The polynomial becomes \(2x^4-3\). The highest power of \(x\) in this remaining polynomial is 4, so its degree is 4. Degree 9 belongs only to the original possible term; it is not counted when its coefficient is zero. Exam tip: remove all zero-coefficient terms before identifying the highest exponent.

The degree of a polynomial is the greatest exponent of its variable in a term with a non-zero coefficient. Here, the powers of y are 6, 3, and 0, so the greatest power is 6. Therefore, the correct answer is 6. The number 14 is a constant term and has degree 0, so it is not the degree of the polynomial. Exam tip: To find the degree, look for the highest power of the variable, not the coefficient.

The governing concept is that degree depends on the variable named in the question, not automatically on every variable appearing in the expression. The question asks for the degree with respect to x. In 9y^4 - 3y^2 + 10, there is no x anywhere, so every term is constant when viewed as a polynomial in x. The expression is therefore an x-polynomial equal to a non-zero constant, and its degree is 0. Option C is correct. Option A is the degree with respect to y, because y^4 is the highest non-zero power of y. Option B incorrectly selects the lower y-power, and option D is unsuitable because the expression is defined and has a valid constant-polynomial degree in x.

The student has confused the number of terms with the degree of a polynomial. The degree is the greatest exponent of the variable whose coefficient is non-zero. In \(7x^5-3x+2\), the exponents of \(x\) are 5, 1, and 0, so its degree is 5. The number 3 only represents the number of terms. Exam tip: identify the non-zero term with the highest power first.

In \(5x^3-2x+7\), the highest power of the variable is 3, so its degree is 3. Option B has degree 4, while C and D are not polynomials because they contain negative and fractional powers respectively. Exam tip: first check that every exponent is a non-negative integer.

First expand: \(x^4(x^2+3)=x^6+3x^4\). Thus, the expression becomes \(x^6+3x^4-x^6+5x^3=3x^4+5x^3\). The \(x^6\) terms cancel. The highest power of \(x\) in the remaining polynomial is 4, so its degree is 4. Although 3 is the power in one term, it is not the highest power. Exam tip: simplify and combine like terms before finding the degree of a polynomial.

In option A, the highest power of \(x\) is 3, and the coefficient of \(x^3\) is 5, which is non-zero. Therefore, its degree is 3. Option B has degree 4, option C has degree 2, and option D is a constant polynomial of degree 0. Exam tip: identify the highest exponent of the variable with a non-zero coefficient.

\((x-4)(x+6)=x^2+2x-24\). The highest power of \(x\) in this polynomial is \(2\), so its degree is \(2\). It is not \(1\), because multiplying the two linear factors produces an \(x^2\) term. Exam tip: simplify the polynomial first, then identify the greatest exponent of the variable.

The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In \(2x^8-6x^3+5\), the highest power of \(x\) is \(8\), so its degree is \(8\). Option A has degree \(7\), while options B and D have degrees \(4\) and \(1\), respectively. Exam tip: To find the degree, look for the highest power of the variable, not the constant term.

A polynomial whose highest power of the variable is 2 has degree 2 and is called a quadratic polynomial. For example, \(3x^2-5x+1\) is quadratic. A linear polynomial has degree 1, so it is not correct. Exam tip: identify the polynomial by its highest exponent.

Combining like terms gives 3x^6-8x^6=-5x^6. Thus, f(x)=-5x^6+4x^2-7. The highest power of x is 6, and its coefficient, -5, is non-zero; therefore, the degree of the polynomial is 6. Option 2 is the power of another term, not the highest power. Exam tip: Always simplify like terms before finding the degree.

The total degree of a polynomial is the greatest sum of the exponents of its variables in any term. For 6x^3y^2, the total degree is 3+2=5; for 5xy, it is 1+1=2; and the constant term -4 has degree 0. Hence, the greatest total degree is 5. Option 6 is incorrect because no term has exponents whose sum is 6. Exam tip: Find the total degree of each term separately, then choose the greatest value.

Find the total degree of each term. For 2x^4y, it is 4+1=5; for -7xy^3, it is 1+3=4; and the constant term 9 has degree 0. The greatest of these is 5, so the total degree of the polynomial is 5. Option 4 is only the total degree of the second term. Exam tip: for a term with more than one variable, add the exponents of all its variables to get its total degree.

To find the degree with respect to \(x\), treat \(y\) as part of the coefficient. The powers of \(x\) in \(5x^4y^3\), \(2xy^6\), and \(-8\) are 4, 1, and 0 respectively. The greatest power is 4, so the correct answer is 4. Although \(y\) has power 6, 6 is not correct because the degree is asked only with respect to \(x\). Exam tip: compare powers only of the variable named in the question.

To find the degree with respect to y, consider only the powers of y. In the given expression, the powers of y are 3, 6, and 0; the greatest is 6, occurring in 2xy^6. Therefore, the correct answer is 6. The value 3 is the power of y only in 5x^4y^3, not the highest power. Exam tip: When degree is asked with respect to one variable, treat the other variables as coefficients.

Terms with zero coefficients disappear and (31) remains. (31) is a non-zero constant polynomial of degree (0).

For degree (3), the (x^4) term must vanish and the (x^3) term must remain. At (t=5), both conditions hold.

The degree of a polynomial is found only after checking whether the coefficient of its highest-looking term is zero. If that coefficient becomes zero, the term disappears, and the next highest non-zero term determines the degree. This makes substitution essential in parameter-based polynomial questions.

Put \(u=-3\) into the expression. The coefficient of \(x^7\) becomes \(u+3=-3+3=0\), so the seventh-degree term vanishes. The coefficient of \(x^5\) becomes \(u^2-2=(-3)^2-2=9-2=7\), which is non-zero. Thus the polynomial reduces to \(7x^5+x^2\). Its highest exponent is \(5\), so option B is correct. Choosing 7 would ignore the zero coefficient of the original leading term.

The degree of a polynomial is not determined by the number of terms. It is the greatest exponent of the variable with a non-zero coefficient. In \(7-2x+x^4\), the exponents of \(7\), \(-2x\), and \(x^4\) are 0, 1, and 4 respectively, so the degree is 4. Option C gives an exponent sum of 5, but adding exponents is not the rule for finding degree. Exam tip: simplify the polynomial, then identify the highest power of the variable.

The polynomial is the product of three linear factors: x, (x-6), and (x+2). Each factor has degree 1, so the degree of the product is 1+1+1=3. On expansion, the highest-power term is x³. Option 6 may result from noticing the constants in the factors, but degree is determined by the highest exponent, not by constants. Exam tip: add the degrees of polynomial factors in a product.

In the given polynomial, \(4x^8-4x^8=0\). Thus, \(g(x)=x^6-x^3\). The highest power with a non-zero coefficient is \(6\), so the degree of the polynomial is \(6\). Option \(8\) is incorrect because the \(x^8\) terms cancel each other. Exam tip: simplify a polynomial by combining like terms before finding its degree.