Degree of a Polynomial

The governing concept is the degree of a polynomial. The degree is the greatest exponent of the variable that has a non-zero coefficient. In 2x^3 - 5x + 1, the terms are 2x^3, -5x, and 1. Their powers of x are 3, 1, and 0 respectively, because a constant term is treated as having exponent zero. The greatest exponent is therefore 3, so option C is correct. The number 2 is the coefficient of x^3, not its exponent. Similarly, 5 is the numerical coefficient of -5x, while 1 is the constant term. Thus none of those numbers represents the highest power. The sign of a coefficient does not affect degree; only the largest surviving exponent matters.

In \(4x^3+x-1\), the highest power of \(x\) is 3, so it is a cubic polynomial, not a quadratic polynomial. \(x^2+3x+2\) is the closest distractor, but its degree is 2, so it is quadratic. \(\sqrt{x}+1\) and \(\frac{1}{x}+2\) are not polynomials because the powers of \(x\) are \(\frac12\) and \(-1\), respectively. Exam tip: in a polynomial, variable exponents must be 0 or positive integers.

A polynomial may have any real coefficients, including fractions, decimals, and irrational numbers. Rewrite the expression as (1/2)x^2 + (-3/4)x + 9. The powers of x are 2, 1, and 0, all of which are non-negative integers, and the coefficient of x^2 is 1/2, which is non-zero. Therefore the expression is a polynomial and its highest power is 2, so its degree is 2. Option A is false because fractional coefficients are allowed. Option C is false because the expression is not identically zero; for example, at x = 0 it equals 9. Option D is false because subtraction is equivalent to adding a negative term, which is permitted in a polynomial.

The degree of a polynomial is the highest power with a non-zero coefficient. The leading coefficient is the coefficient of that highest-degree term. In option B, the highest power is x^2, so the degree is 2, and the coefficient of x^2 is -4, which is negative. Thus both conditions are satisfied. Option A also has degree 2, but its leading coefficient is +3. Option C has a negative leading coefficient, -2, but its degree is 3 because of the x^3 term. Option D is linear, with degree 1, and its leading coefficient is +5; its negative constant term does not affect the leading coefficient. Therefore option B is the only complete match.

A polynomial can have a missing intermediate power; the missing term is understood to have coefficient zero. In option A, 2x^3 + 5x - 1 has highest power x^3 with non-zero coefficient 2, so its degree is 3. There is no x^2 term, which means its x^2 coefficient is 0. Hence A satisfies both requirements. Options B and D each contain x^2 with coefficient 1, so they fail the second condition. Option C has no x^3 term and its highest power is x^2, so its degree is only 2. The absence of x^2 does not reduce the degree when a higher non-zero x^3 term is present.

A coefficient is the numerical factor attached to a specified power of the variable. When a power is absent from a polynomial, its coefficient is understood to be zero. The expression 7x⁵−4x³+8 contains the terms 7x⁵, −4x³ and 8x⁰. It has no x⁴ term, no x² term and no x term. Therefore the requested coefficients, in the stated order, are 0, 0 and 0, making option B correct. Option A lists coefficients that belong to existing terms but assigns them to the wrong powers. Option C mistakes exponents for coefficients. Option D incorrectly transfers the coefficient −4 of x³ to x⁴ and the constant 8 to x. Missing terms are conventionally represented with zero coefficients, which is essential when comparing or adding polynomials.

In \(x^4-2x+6\), the highest power of \(x\) is \(4\), and all exponents of \(x\) are non-negative integers. Therefore, it is a polynomial in \(x\) of degree \(4\). \(x^3+4\) is a polynomial, but its degree is \(3\). In \(\frac{1}{x}\) and \(\sqrt{x}\), the exponents of \(x\) are \(-1\) and \(\frac12\), so they are not polynomials. Exam tip: exponents of a variable in a polynomial can only be \(0,1,2,\ldots\).

In the polynomial \(5x^3-2x+7\), the powers of \(x\) are 3, 1, and 0 respectively. The greatest power is 3, so the degree of the polynomial is 3. The constant term \(7\) has degree 0. Exam tip: To find the degree of a polynomial, identify the highest power of the variable.

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. The terms here are 8x^2, 4x, and -9. The greatest exponent of x is 2, so the polynomial has degree 2. The number 8 is a coefficient, not the degree. Exam tip: First simplify the polynomial, then identify the highest power of its variable.

The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In \(6x-13\), the exponent of \(x\) is \(1\), so its degree is \(1\). The number \(6\) is only the coefficient of \(x\), not the degree. Exam tip: To find degree, look for the highest power of the variable, not the coefficient or constant term.

A non-zero constant polynomial has degree \(0\). We can write \(-18=-18x^0\), so the highest exponent of \(x\) is \(0\). A polynomial of degree \(1\) must contain a variable with exponent \(1\), such as \(3x-18\). Exam tip: Only the zero polynomial has an undefined degree; a non-zero constant does not.

The coefficient of \(0x^5\) is zero, so this term does not contribute to the degree of the polynomial. In the remaining polynomial \(7x^4-3x+2\), the highest power of \(x\) is 4; therefore, its degree is 4. Option 5 is incorrect because the \(x^5\) term is actually zero. Exam tip: Remove terms with zero coefficients before finding the degree.

The degree of a polynomial is the greatest exponent of its variable. Here, the exponents of the terms are 6, 2, and 0, and the greatest is 6. Therefore, the degree of the polynomial is 6. The number 2 is only the exponent of the term x^2, not of the whole polynomial. Exam tip: treat a non-zero constant term as having exponent 0.

The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. In the given polynomial, the highest power of x is 4 in the term x^4. Therefore, its degree is 4. The term x^3 has exponent 3, so it cannot determine the degree. Exam tip: Simplify a polynomial first and ignore terms with zero coefficients before finding its degree.

In the polynomial \(9x^2\), the exponent of the variable \(x\) is 2. The degree of a polynomial in one variable is the greatest exponent of that variable, so its degree is 2. The number 9 is the coefficient, not the degree. Exam tip: look at the exponent of the variable, not the coefficient.

The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In \(2x^3+6\), the highest power of \(x\) is \(3\), so its degree is \(3\). \(4x^2+1\) is the closest distractor, but its degree is \(2\). Exam tip: A non-zero constant polynomial has degree \(0\).

The degree of a non-zero polynomial is the greatest exponent of the variable whose coefficient is non-zero. In 4 + 3x² − 2x⁵, the constant term 4 can be viewed as 4x⁰, so its degree is 0. The term 3x² has degree 2, and the term −2x⁵ has degree 5. Because the coefficient of x⁵ is −2, not zero, the greatest relevant exponent is 5. Therefore the degree of the polynomial is 5, making option D correct. The order of the terms does not affect the degree. Options 2, 3, and 4 are distractors based on selecting a smaller exponent or the numerical coefficient, but degree depends only on the highest non-zero exponent of the variable.

The degree of a polynomial is the highest exponent of its variable with a non-zero coefficient. Here, the exponents of the terms are 3, 2, and 0, and the greatest is 3. Therefore, the degree is 3. Option 2 is only the degree of the term 2x^2, not of the whole polynomial. Exam tip: a non-zero constant term such as -7 has degree 0.

In \(9x-5\), the variable term is \(9x=9x^1\). The highest exponent of \(x\) is \(1\), so the degree of the polynomial is \(1\). The constant term \(-5\) has degree \(0\), so it does not determine the degree. Exam tip: find the highest power of the variable to identify the degree of a polynomial.

The degree of a polynomial is the greatest exponent of the variable whose coefficient is non-zero. Here the terms are 6x^5, -3x^2, x, and 8, and the greatest exponent of x is 5. Therefore, the degree of the polynomial is 5. The constant term 8 has degree 0; it does not make the degree 8. Exam tip: To find the degree, look for the highest power of the variable, not the numerical coefficient.

The non-zero constant polynomial 12 has no variable term. It can be written as \(12x^0\), so the highest power of the variable is 0. Therefore, its degree is 0. The number 12 is its constant value, not its degree. Exam tip: Every non-zero constant has degree 0; only the zero polynomial is generally treated as having an undefined degree.

The coefficient of 0x^4 is 0, so it does not count as a term of the polynomial. The remaining polynomial is 7x^2-3x+1, whose highest power of x is 2. Therefore, its degree is 2. Choosing 4 is incorrect because the coefficient of x^4 is zero. Exam tip: Remove terms with zero coefficients before finding the degree.

Option A is correct. The degree of a polynomial is determined by the term containing the greatest power of the variable, provided that the coefficient of that term is not zero. In 3x⁴ + x³ + 5, the powers of x are 4, 3, and 0 respectively. The term 3x⁴ contains the highest power, x⁴, so it determines the degree and shows that the polynomial has degree 4. The number 3 is only the coefficient of x⁴; it is not the degree. The term x³ has a lower power, and 5 is a constant term whose degree is 0. Therefore option A names the correct determining term, while the degree itself is 4.

The degree of a polynomial is the greatest exponent of its variable. The terms are x^2, x^7, and 4x, whose exponents of x are 2, 7, and 1 respectively. The greatest exponent is 7, so the degree is 7. The number 4 is a coefficient, not an exponent. Exam tip: changing the order of terms does not change the degree of a polynomial.

Combining like terms gives 5x^3-2x^3=3x^3. Thus, the simplified polynomial is 3x^3+6x-1. The highest exponent of x is 3, so its degree is 3. In 6x, 6 is a coefficient, not an exponent; therefore, 6 cannot be the degree. Exam tip: Simplify a polynomial first, especially when leading terms may combine or cancel, before finding its degree.