Polynomials in one variable

Terms are separated by plus or minus signs. The terms here are \\(4x^2\\), \\(-6x\\), and \\(1\\), so there are 3 terms and the polynomial is a trinomial. In an exam, count the negative sign with its term; \\(-6x\\) is one term, not two.

Substituting 1 for \(x\) gives \(g(1)=3(1)^2-1=3-1=2\). Therefore, option B is correct. Remember to evaluate \(x^2\) before performing the subtraction; taking only \(3x^2\) would incorrectly give 3.

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. In option B, the highest-degree term is \(-2x^3\), so its leading coefficient is \(-2\). The leading coefficients of the other options are 5, 1, and 3. Exam tip: identify the highest power first, then read its coefficient.

The statement is incorrect. \(3x^2-5x+7\) has only one variable, \(x\), with powers 2, 1 and 0, all non-negative integers. The number of terms does not decide the number of variables. Exam tip: a constant has power 0.

The governing concept is the leading term of a polynomial. When a polynomial is arranged in descending powers, its leading term is the complete term having the greatest power of the variable. The coefficient and its sign are both part of that term. In 2x^2 + 5x - 3, the powers of x are 2, 1, and 0. Since 2 is the greatest exponent, the first term, 2x^2, is the leading term. Therefore option A is correct. The term 5x has lower degree 1, while -3 is the constant term with degree 0. Option D is incomplete because x^2 gives only the variable part and leaves out the coefficient 2, so it is not the complete leading term.

The governing concept is the definition of a constant polynomial. A constant polynomial has a fixed value and contains no variable term; its degree is 0 for a non-zero constant. The expressions 8, -3, and 0 are all constants, so their values do not change with x. In contrast, 2x + 1 contains the variable x and changes when x changes, so it is a linear polynomial rather than a constant polynomial. Therefore option D is correct. The zero polynomial is also treated as a constant polynomial in this classification, although its degree is often handled separately or described as undefined. The presence of x, not the number of terms, is the decisive distinction here.

The governing concept is identifying separate terms in a polynomial. Terms are separated by plus or minus signs, and a sign belongs to the term that follows it. Therefore, x² - 5x + 6 has the three terms x², -5x, and 6, so option A is correct. Option B loses the negative sign, C separates coefficients from variables, and D groups unlike parts together.

The governing concept is substitution of known parameter values into a polynomial and then simplifying. Begin with p(x) = x^3 + ax^2 + bx + 1. Replacing a by 0 gives ax^2 = 0x^2 = 0, so the quadratic term disappears completely. Replacing b by -2 gives bx = -2x. Hence p(x) = x^3 + 0x^2 - 2x + 1, which simplifies to x^3 - 2x + 1. Therefore option A is correct. Option B changes the negative value of b into a positive one. Option C attaches -2 to x^2 instead of x, and option D both mishandles the substitution and omits the constant term 1. Careful matching of each parameter with its term avoids these errors.

The governing concept is the definition of a zero of a polynomial. A number a is a zero of p(x) exactly when p(a) = 0. Substitute x = −3 into each option. For A, p(−3) = −3 + 3 = 0, so −3 is a zero of x + 3. For B, the value is −3 − 3 = −6. For C, it is 3(−3) + 1 = −9 + 1 = −8. For D, it is (−3)^2 + 3 = 9 + 3 = 12. Since only option A produces zero, A is correct. The plus sign in x + 3 is essential: it cancels the negative input. Direct substitution is more reliable than choosing by visual appearance alone.

The governing concept is identification of coefficients and the constant term in a polynomial. In a term such as −7x², the coefficient of x² is −7 because it multiplies x². The constant term is the term containing no variable, so in this polynomial it is −11. The question asks for their algebraic sum, not their product or their separate values. Therefore, (−7) + (−11) = −18. Option A is correct. Option B gives only the constant term, and option C gives only the coefficient of x². Option D does not result from combining the requested signed values. The negative signs must be retained throughout the addition; adding the absolute values would give an incorrect result.

The degree of a nonzero polynomial is the greatest exponent of x whose coefficient is nonzero. Since a=0, the cubic term ax³ disappears completely. The next term is bx², and b≠0 guarantees that this term remains present. Therefore the highest surviving power is x², so degree(p)=2 and option C is correct. The value of c or d does not affect the degree once a nonzero x² term is already present. Option D incorrectly retains the vanished cubic term. Option B would be possible only if b were also zero and c were nonzero, while option A would describe a nonzero constant polynomial, not the general expression under the stated condition. The condition b≠0 is essential because it prevents further reduction of the degree.

A number r is a zero of a polynomial p(x) when p(r)=0. To test x=0, substitute zero into each option. For x³−2x, the value is 0−0=0. For 5x²+x, it is 0+0=0. For x⁴, it is 0, so all three have zero as a zero. For x²+1, the value is 0²+1=1, not zero; therefore option D is the unique correct answer. The decisive feature is the constant term: at x=0, every positive-power term vanishes and the polynomial value equals its constant term. The first three expressions have constant term 0, whereas x²+1 has constant term 1.

The governing concept is the definition of a zero of a polynomial: a number r is a zero of p(x) exactly when p(r)=0. Substitute x=0 into every option. For A, p(0)=0^2+5=5, so it is not zero. For B, p(0)=3(0)^3-2(0)=0, so it satisfies the required condition. For C, p(0)=0^4+1=1, and for D, p(0)=2(0)^2-7(0)+9=9. Thus only option B is correct. The factor theorem gives the same result because 3x^3-2x=x(3x^2-2), so x is a factor and x=0 is a zero. The other polynomials have nonzero constant terms, which prevents their value at zero from being zero.

The governing concept is the degree of a nonzero polynomial. A linear polynomial has degree exactly 1, meaning the highest power of the variable with a nonzero coefficient is x^1. In option A, 5x-9 has highest exponent 1 and therefore is linear. Option B, x^2-9, has degree 2 and is quadratic. Option C is a nonzero constant polynomial, whose degree is 0. Option D, x^3+x, has highest exponent 3 and is cubic, even though it also contains a linear term. Thus option A is the only correct answer. The classification depends on the greatest exponent, not simply on whether the expression contains x or has only a few terms. The nonzero coefficient of x in 5x-9 confirms its degree is exactly one.

The governing concept is the definition of a zero of a polynomial: a number r is a zero when p(r)=0. The polynomial can be factored as p(x)=7x^3-2x=x(7x^2-2). Thus x is a factor of every term, and substituting x=0 gives p(0)=0(7(0)^2-2)=0. Equivalently, the constant term is zero, which is why the graph passes through the origin. Option A states the direct algebraic reason. Option B is incorrect because 7 is the coefficient of x^3, not the constant term. The degree being 3 and the presence of a negative coefficient do not by themselves make zero a root. Therefore A is the only unambiguous answer.