Polynomials in one variable

In the standard form of a polynomial, its terms are arranged in descending order of their powers. In option A, the powers are 3, 2, 1 and 0, so it is in standard form. Options B, C and D contain the same terms but are not arranged in descending order. Exam tip: begin with the term having the highest power and then write the remaining terms in decreasing powers.

The polynomial contains \(x^4\), \(3x^2\), \(x\), and the constant term \(2\), but it does not contain an \(x^3\)-term. Therefore, the coefficient of \(x^3\) is considered to be 0. Exam tip: Arrange the terms in descending powers to identify a missing power quickly.

The governing concept is the constant term of a polynomial. A constant term is independent of the variable; equivalently, it is the coefficient of x⁰. The expression p(x) = 7x² has only the term 7x² and contains no separately written term without x. We may therefore write it as 7x² + 0, so the constant term is 0 and option C is correct. The number 7 is the coefficient of x², not the constant term, while 2 is the exponent of x. The number 1 has no role in the expression. A useful check is to put x = 0: p(0) = 7(0)² = 0. For a polynomial, its value at zero equals its constant term, confirming the answer.

Substituting \(x=-2\), we get \(p(-2)=(-2)^2+4(-2)+4=4-8+4=0\). Therefore, the correct answer is 0. The value 4 is only \((-2)^2\), not the value of the complete polynomial. In exams, use brackets carefully when substituting a negative number.

Degree is decided by the highest power (2), not by the number of terms. Avoid this common exam mistake.

There is no (x^2)-term in this polynomial, so its coefficient is (0). The coefficient of a missing term is always taken as (0).

To find \(p(1)\), substitute 1 for every \(x\) in the polynomial: \(p(1)=1^2+3\times1+2=1+3+2=6\). Therefore, the correct answer is 6. The nearby option 5 may result from adding the terms incorrectly or omitting a term. Exam tip: substitute the given value for each occurrence of the variable before simplifying.

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.