Polynomials in one variable

The degree of a polynomial is the highest exponent of its variable. In this polynomial, the powers of \(x\) are 3, 1, and 0, so its degree is 3. The number 9 is the constant term, not the degree. Exam tip: identify the greatest exponent of the variable.

In the polynomial \\(5x^2-8x+6\\), the term containing \\(x^2\\) is \\(5x^2\\), so its coefficient is 5. The number -8 is the coefficient of \\(x\\), while 6 is the constant term. Exam tip: identify the numerical multiplier attached to the required variable term.

The constant term is the term that does not contain the variable \(x\). In this polynomial, \(2x^2\) and \(3x\) contain \(x\), whereas \(-11\) does not; therefore, \(-11\) is the correct answer. Exam tip: the constant term can be viewed as the coefficient of \(x^0\).

The terms of this polynomial are \(4x^3\), \(x^2\), \(-7x\), and \(1\), so there are 4 terms in total. Terms are separated by plus or minus signs; \(-7x\), including its negative sign, is one term. In an exam, remember to count the constant term \(1\) as well.

A polynomial is called constant when it has no variable term. The expression 9 is simply a fixed number, so its value remains 9 regardless of the value assigned to x. Therefore it is a constant polynomial, and choice C is correct. It is not linear, quadratic, or cubic because those types require a variable with highest powers 1, 2, or 3 respectively.

For a non-zero constant polynomial, the degree is 0. This is because 9 can be written as 9x^0, and the highest exponent present is 0. The number 9 is not the degree; it is the coefficient or value of the polynomial. The special zero polynomial needs separate treatment, but that issue does not arise here because 9 is non-zero.

The degree of a polynomial is the highest exponent of its variable. In \(7x^2-3x+5\), the highest exponent is 2, so it is quadratic. A constant term or a negative coefficient is allowed. Exam tip: identify the highest power first.

Riya is correct. In \(5x^3-2x+7\), the exponents of \(x\) are 3, 1, and 0; the greatest is 3, so the degree is 3. Exam tip: count exponents, not terms.

(\frac{3}{x}) means (3x^{-1}), which is not allowed in a polynomial. Powers of the variable must be whole numbers.

Substituting x=1 in the polynomial gives (p(1)=1^2+2(1)+1=1+2+1=4). Therefore, the correct answer is 4. Exam tip: replace x in every term, not just in one term; doing so incorrectly can lead to a distractor such as 3.

To find p(2), substitute 2 for x: p(2) = 3(2) - 5 = 6 - 5 = 1. Therefore, the correct answer is 1. Exam tip: For the value of a polynomial at a given number, substitute that number directly for the variable.

Substituting \(x=0\) in the polynomial gives \(q(0)=0^3-0=0-0=0\). Therefore, the correct answer is B, 0. Do not confuse \(0^3\) with 1; every positive power of zero is 0. Exam tip: To evaluate a polynomial, substitute the given value at every occurrence of \(x\).

The governing concept is that the degree of a non-zero polynomial is the largest exponent of the variable with a non-zero coefficient. In 6x^4 − 2x^2 + x − 8, the powers of x are 4, 2, 1 and 0. The coefficient of x^4 is 6, which is non-zero, so the leading term is 6x^4. Consequently, the degree of the polynomial is 4 and option C is correct. The number 2 is the exponent of the second term but not the largest exponent; 1 is the exponent of x; and 8 belongs to the constant term, not the degree. A negative coefficient does not affect the degree, and lower-power terms cannot override the highest non-zero power. Thus examining the leading term gives the answer immediately.

In the standard form ax² + bx + c, the coefficient of x is b. Here, the x-term is −7x, so its coefficient is −7. Option 7 is incorrect because it omits the negative sign. In an exam, include the sign attached to the term when identifying its coefficient.

(\sqrt{2}) is a real number and the powers of the variable are whole numbers. Real coefficients are allowed in polynomials.

In (x^{\frac{1}{2}}), the power of the variable is not a whole number. In a polynomial, powers must be (0,1,2,\ldots).

There is no (x^3) term, so its coefficient is (0). The coefficient of a missing term is taken as (0).

The governing concept is the degree of a polynomial, defined as the greatest exponent of x having a nonzero coefficient. The zero polynomial has every coefficient equal to zero, so it has no nonzero term and therefore no greatest exponent under the usual school-level definition. Its degree is consequently not defined. This must be distinguished from a nonzero constant polynomial such as 5, whose only nonzero term is 5x⁰ and whose degree is 0. Thus option A is incorrect because it confuses the zero polynomial with a nonzero constant; options B and D assign exponents without a corresponding nonzero term. Option C correctly states the standard result.

Substituting \(x=-1\), we get \(p(-1)=2(-1)^2-3(-1)+4=2+3+4=9\). Hence, 9 is correct. Remember that \((-1)^2=1\) and \(-3(-1)=+3\); treating either sign incorrectly can lead to distractor values. Exam tip: Always place a negative substituted value in brackets.

The polynomial is arranged in descending powers of \(x\). The highest power is 4, so the term \(x^4\) is the leading term. Option C, \(2x^3\), has only degree 3 and therefore cannot be the leading term. In an exam, identify the term containing the highest power of the variable.

The leading term is (3x^2), so the leading coefficient is (3). The coefficient of the highest degree term is the leading coefficient.

A term is counted separately when it is written as an algebraic part of the polynomial. In 4x² + 0x + 9, the middle term 0x is explicitly present, and its coefficient is 0. Therefore exactly one written term has a zero coefficient. The term 4x² has coefficient 4, and the constant term 9 has coefficient 9, so neither qualifies. Although 0x contributes nothing to the numerical value and the expression can be simplified to 4x² + 9, the question asks what is explicitly written before simplification. Thus the correct count is 1. The answer is not 0 because 0x is visible, and it is not 2 or 3 because the other written terms have non-zero coefficients.

The direct answer is C, zero. A number a is called a zero of a polynomial p(x) when substituting x=a makes the polynomial equal to 0; in symbols, \(p(a)=0\). The given statement already has exactly this definition. Option C is therefore correct. Option A, coefficient, is a number multiplying a term, such as 5 in \(5x\). Option B, degree, is the highest power of the variable, such as 2 in \(x^2+1\). Option D, term, is one part separated by plus or minus signs, such as \(x^2\) or 3. None of these definitions means that the value of the polynomial is zero. Memory cue: substitute a; if the result is zero, a is a zero.

In \(\frac{5}{x^2}+1=5x^{-2}+1\), the power of \(x\) is \(-2\). In a polynomial, the powers of the variable must be non-negative integers, so this expression is not a polynomial. Option D, \(7x+8\), is a linear polynomial because the power of \(x\) is 1. Exam tip: when the variable appears in the denominator, its power becomes negative, which generally disqualifies the expression from being a polynomial.

To find the zero, set \(p(x)=0\): \(x-6=0\), which gives \(x=6\). Therefore, the correct answer is 6. Choosing -6 results from an incorrect sign; the zero of \(x-a\) is \(a\). Exam tip: Always set the polynomial equal to zero when finding its zero.

There is no (x^2) term, so the coefficient is (0). A missing power can be treated as a term with zero coefficient.