Polynomials in one variable

Option B, 5x³ − 4, contains two distinct terms: 5x³ and −4. Therefore, it is a binomial. Option A has three terms, option C has one term, and option D has four terms. In an exam, count each part separated by a plus or minus sign as one term.

The term containing \(x\) is \(-x\), which can be written as \(-1x\). Therefore, the coefficient of \(x\) is \(-1\). Option C misses the negative sign. Exam tip: when identifying a coefficient, include both the numerical multiplier and its sign.

The degree of a polynomial is the highest power of its variable. In \\(x^5+1\\), the highest power of \\(x\\) is 5, so its degree is 5. The term 1 is a constant and does not increase the degree. In an exam, identify the maximum exponent of the variable rather than adding the exponents.

In (y^2-3y+2), the powers of (y) are whole numbers. A polynomial can be in one variable such as (y).

\(r\left(\frac{1}{2}\right)=4\left(\frac{1}{2}\right)^2=1\). Use brackets when substituting fractions.

Terms in a polynomial are separated by plus or minus signs. Here, \(7x^2\) and \(5\) are two distinct terms, so the polynomial has 2 terms. In an exam, count each part separated by a plus or minus sign as one term.

Since \(a\neq 0\), the term \(ax^2\) is non-zero, so the highest power of \(x\) is 2. Therefore, the polynomial has degree 2 and is a quadratic polynomial. A linear polynomial has degree 1, while a cubic polynomial has degree 3. Exam tip: Identify the polynomial by its highest power of the variable.

In a polynomial in one variable, powers of the variable are non-negative integers. In exams, reject negative powers and roots of the variable.

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. Here, the highest exponent is 5 and its coefficient is 4, so the degree is 5. The term 9 is a constant term and does not determine the degree. In an exam, identify the greatest exponent of the variable.

The term containing \(x^2\) is \(2x^2\), so its coefficient is 2. The number 7 is the coefficient of \(x^3\), while -1 is the coefficient of \(x\); neither is the coefficient of \(x^2\). In an exam, match the required power with its term and then read its numerical factor.

A constant term is a term that does not contain the variable \(x\). In the given polynomial, \(5x^4\) and \(-8x\) contain \(x\), whereas 11 does not; therefore, 11 is the constant term. The 4 in option D is the exponent, not a term. Exam tip: the standalone number in a polynomial is usually its constant term.

The terms of the polynomial are \(2x^3\), \(-5x^2\), \(4x\), and \(-1\), so it has 4 terms. Terms are separated by addition or subtraction signs; do not count coefficients or powers separately. Exam tip: count each positive or negative term individually.

To find \(p(2)\), substitute 2 for \(x\): \(p(2)=3(2)^2-2(2)+5=12-4+5=13\). Therefore, the correct answer is 13. The distractor 11 may result from combining the terms incorrectly. Exam tip: evaluate the exponent first, then perform multiplication and addition or subtraction.

Substituting \(x=-1\), we get \(q(-1)=(-1)^3-4(-1)^2+(-1)+6=-1-4-1+6=0\). Here, \((-1)^2=1\) but \((-1)^3=-1\), so the signs must be handled carefully. In an exam, evaluate each term separately before adding them.

The polynomial has three distinct terms: \\(8x^2\\), \\(-3x\\), and \\(1\\). Therefore, it is a trinomial. A binomial has exactly two terms, while a zero polynomial has all its coefficients equal to zero. In an exam, count the terms to classify a polynomial.

The governing concept is classification by polynomial degree. A polynomial is quadratic exactly when the highest power of its variable with a non-zero coefficient is 2. In p(x) = ax² + bx + c, the coefficient of x² is a. Therefore a must be non-zero, so a ≠ 0 is the required condition and option B is correct. If a = 0, the x² term disappears and the expression becomes bx + c, which may be linear or constant. The values b = 0 or c = 0 are not necessary: for example, 2x² + 0x + 3 is still quadratic, and 2x² + 5x + 0 is also quadratic. Thus the other options do not correctly state the defining condition.

A polynomial whose all coefficients are zero is called the zero polynomial; therefore, \(p(x)=0\) is the zero polynomial. It is not classified as a non-zero constant polynomial, and its degree is considered undefined. Exam tip: remember that the degree of the zero polynomial is undefined.

A non-zero constant polynomial contains no variable term, so it can be written as \(12x^0\); therefore, its degree is 0. The value 12 is the constant term, not the degree. Remember that the degree of the zero polynomial is generally undefined, but this polynomial is non-zero.

The term (0x^3) is ineffective, and the highest non-zero power is (4). Ignore terms with zero coefficient.

In \(4x^3-2x+7\), the powers of \(x\) are 3, 1 and 0, all non-negative integers; hence it is a polynomial in one variable. \(3/x=3x^{-1}\) has a negative exponent. Exam tip: check that every variable exponent is 0, 1, 2, ... .