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339 questions tagged with definition-polynomial.
Which option is not a polynomial in (x), even though an \(x^2\) term is present?
B. \(x^2+\sqrt{x}+4\)
\(x^2\) पद होना पर्याप्त नहीं है। \(\sqrt{x}=x^{1/2}\) अमान्य घात वाला पद है। / Having an \(x^2\) term is not enough. \(\sqrt{x}=x^{1/2}\) is a term with an invalid power.
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If (p(x)=\lambda x-2+4x-3) has degree (2), what is the correct condition for \(\lambda\)?
B. \(\lambda\neq0\)
घात (2) तभी होगी जब \(x^2\) का गुणांक शून्य न हो। इसलिए \(\lambda\neq0\) चाहिए। / The degree will be (2) only if the coefficient of \(x^2\) is non-zero. Therefore, \(\lambda\neq0\) is required.
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Which expression is a polynomial in (x) but has only even powers of (x)?
A. \(x^6-3x^4+2x^2-8\)
पहले विकल्प में घातें (6), (4), (2) और (0) हैं। ये सभी सम और बहुपद के लिए मान्य हैं। / In the first option, the powers are (6), (4), (2), and (0). All are even and valid for a polynomial.
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If (p(x)=(s-2)x-3+(s+1)x-2+5) is a non-zero constant polynomial, what is possible?
C. कोई ऐसा (s) नहीं हैNo such (s) exists
स्थिर बहुपद के लिए (s-2=0) और (s+1=0) दोनों चाहिए। एक ही (s) दोनों शर्तें पूरी नहीं कर सकता। / For a constant polynomial, both (s-2=0) and (s+1=0) are required. One value of (s) cannot satisfy both conditions.
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Which option states the definition of a polynomial most correctly?
A. चर की घातें शून्य या धनात्मक पूर्णांक हों और गुणांक वास्तविक हो सकते हैंVariable powers are zero or positive integers and coefficients may be real
बहुपद की मुख्य शर्त चर की घातों पर होती है। घातें शून्य या धनात्मक पूर्णांक होनी चाहिए। / The main condition for a polynomial is about the powers of the variable. The powers must be zero or positive integers.
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What is the degree of (p(x)=0x-3+0x-2+0x+6)?
D. (0)
यह शून्य से अलग स्थिर बहुपद (6) है। ऐसे बहुपद की घात (0) होती है। / This is the non-zero constant polynomial (6). Such a polynomial has degree (0).
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Which expression is a polynomial but does not contain all consecutive powers?
A. \(x^5+2x^2-3\)
बहुपद में बीच की घातों का होना जरूरी नहीं है। \(x^5+2x^2-3\) में उपस्थित घातें (5), (2), (0) हैं और मान्य हैं। / A polynomial need not contain all middle powers. In \(x^5+2x^2-3\), the present powers (5), (2), (0) are valid.
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If \(a\neq0\), what can definitely be said about \(ax^4+bx^2+c\)?
D. यह घात (4) का बहुपद हैIt is a polynomial of degree (4)
\(a\neq0\) होने से \(x^4\) पद मौजूद रहता है। इसलिए सबसे बड़ी घात (4) निश्चित है। / Since \(a\neq0\), the \(x^4\) term remains present. Therefore, the highest power is definitely (4).
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Which expression is a polynomial with real coefficients and an irrational constant term?
D. \(3x^3-2x+\sqrt{11}\)
\(\sqrt{11}\) स्थिर वास्तविक संख्या है और मान्य है। चर की सभी घातें शून्य या धनात्मक पूर्णांक हैं। / \(\sqrt{11}\) is a real constant and is allowed. All powers of the variable are non-negative integers.
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Which expression is not a polynomial because it contains \(\frac{1}{x^4}\)?
A. \(x^4+\frac{1}{x^4}\)
\(\frac{1}{x^4}=x^{-4}\) है, जो ऋणात्मक घात है। बहुपद में ऋणात्मक घातें स्वीकार्य नहीं हैं। / \(\frac{1}{x^4}=x^{-4}\), which is a negative power. Negative powers are not allowed in polynomials.
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If (p(x)=kx-2+3x+4) has degree (1), what should be the value of (k)?
A. (0)
घात (1) के लिए \(x^2\) का गुणांक शून्य होना चाहिए। (k=0) रखने पर (3x+4) रैखिक बहुपद बचता है। / For degree (1), the coefficient of \(x^2\) must be zero. With (k=0), (3x+4) remains a linear polynomial.
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Which statement is correct about \(x^0+x^{-1}+2\)?
C. यह बहुपद नहीं है क्योंकि \(x^{-1}\) हैIt is not a polynomial because it has \(x^{-1}\)
\(x^0\) मान्य है लेकिन \(x^{-1}\) मान्य नहीं है। एक अमान्य पद पूरे व्यंजक को बहुपद नहीं रहने देता। / \(x^0\) is valid but \(x^{-1}\) is not valid. One invalid term prevents the whole expression from being a polynomial.
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Which is the correct identification of (p(x)=7x-3-4x-2+0x-5+11)?
C. घात (3) का बहुपदPolynomial of degree (3)
\(0x^5\) शून्य पद है और घात नहीं बढ़ाता। सबसे बड़ी अशून्य घात (3) है। / \(0x^5\) is a zero term and does not raise the degree. The highest non-zero power is (3).
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B. \(u^2+\frac{1}{x}\)
(u) के संदर्भ में \(\frac{1}{x}\) स्थिर जैसा है, इसलिए यह (u) में बहुपद है। लेकिन (x) के संदर्भ में \(\frac{1}{x}\) की घात (-1) है। / With respect to (u), \(\frac{1}{x}\) is like a constant, so it is a polynomial in (u). With respect to (x), \(\frac{1}{x}\) has power (-1).
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If (p(x)=\(a^2-9\)x-4+2x+1) is a linear polynomial, which values of (a) are possible?
C. (a=3) या (a=-3)(a=3) or (a=-3)
रैखिक होने के लिए \(x^4\) का गुणांक शून्य होना चाहिए और (2x) पद मौजूद है। \(a^2-9=0\) से \(a=\pm3\)। / For it to be linear, the \(x^4\) coefficient must be zero and the (2x) term is present. From \(a^2-9=0\), \(a=\pm3\).
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Which polynomial has four terms and degree (5)?
A. \(x^5+x^3-x+2\)
\(x^5+x^3-x+2\) में (4) पद हैं और सबसे बड़ी घात (5) है। हर में चर नहीं है। / \(x^5+x^3-x+2\) has (4) terms and the highest power is (5). There is no variable in the denominator.
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Which statement is correct about \(x^2+\frac{x}{2}+9\)?
B. यह बहुपद है और घात (2) हैIt is a polynomial and has degree (2)
\(\frac{x}{2}\) का अर्थ \(\frac{1}{2}x\) है, जहाँ घात (1) है। सभी घातें मान्य हैं। / \(\frac{x}{2}\) means \(\frac{1}{2}x\), where the power is (1). All powers are valid.
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If (q(x)=(c+5)x-3+(c-2)x-2+4x-1) must have degree (2), what condition is needed?
C. (c=-5)
घात (2) के लिए \(x^3\) का गुणांक शून्य होना चाहिए। (c+5=0) से (c=-5) और \(x^2\) का गुणांक (-7) अशून्य रहता है। / For degree (2), the coefficient of \(x^3\) must be zero. From (c+5=0), (c=-5), and the \(x^2\) coefficient (-7) remains non-zero.
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In which option are the polynomial and its degree both correct?
A. \(5x^6-2x^3+1\), घात (6)\(5x^6-2x^3+1\), degree (6)
पहले विकल्प में सभी घातें शून्य या धनात्मक पूर्णांक हैं और सबसे बड़ी घात (6) है। बाकी विकल्प बहुपद नहीं हैं। / In the first option, all powers are non-negative integers and the highest power is (6). The other options are not polynomials.
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Which expression is not a polynomial in (x) because it has a fractional power?
D. (x^{52}+x+1)
\(x^{5/2}\) में घात \(\frac{5}{2}\) है जो पूर्णांक नहीं है। बहुपद में ऐसी घात मान्य नहीं होती। / In \(x^{5/2}\), the power is \(\frac{5}{2}\), which is not an integer. Such a power is not valid in a polynomial.
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After simplifying \(\frac{x^3+2x}{x}\), what kind of expression will it become?
C. द्विघात बहुपदQuadratic polynomial
सरल करने पर \(\frac{x^3+2x}{x}=x^2+2\) मिलता है। यह घात (2) का बहुपद है। / On simplifying, \(\frac{x^3+2x}{x}=x^2+2\). It is a polynomial of degree (2).
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If (p(x)=(r-1)x-2+(r+2)x+5) is a constant polynomial, what is the correct conclusion about (r)?
B. ऐसा कोई (r) नहीं हैNo such (r) exists
स्थिर बहुपद के लिए (r-1=0) और (r+2=0) दोनों चाहिए। ये दोनों एक साथ संभव नहीं हैं। / For a constant polynomial, both (r-1=0) and (r+2=0) are required. These cannot hold together.
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Which expression is a polynomial in (x) if (m) is treated as a constant?
D. \(m^2x^3-5mx+6\)
(m) को (x) के लिए गुणांक माना जा सकता है। (x) की घातें (3), (1) और (0) हैं। / (m) can be treated as a coefficient with respect to (x). The powers of (x) are (3), (1), and (0).
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If (p(x)=bx-4+3x-3-2x+1) and (b=0), what will be the degree of (p(x))?
C. (3)
(b=0) होने पर \(x^4\) वाला पद हट जाता है। तब सबसे बड़ी घात (3) रहती है। / When (b=0), the \(x^4\) term disappears. Then the highest power remains (3).
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Which option is exactly a cubic polynomial in (y)?
A. \(4y^3-2y+8\)
ठीक घन बहुपद की सबसे बड़ी घात (3) होती है। \(4y^3-2y+8\) में सबसे बड़ी घात (3) है। / An exactly cubic polynomial has highest power (3). In \(4y^3-2y+8\), the highest power is (3).
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Which statement is correct about \(x^2+\sqrt{x}+1\)?
B. यह बहुपद नहीं है क्योंकि (\sqrt{x}=x^{12}) है / It is not a polynomial because \(\sqrt{x}=x^{1 / 2}\)
बहुपद में चर की घातें पूर्णांक होनी चाहिए। \(\frac{1}{2}\) पूर्णांक नहीं है। / In a polynomial, powers of the variable must be integers. \(\frac{1}{2}\) is not an integer.
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Under which condition will ((k+2)x-3+(5-k)x-2+7) have degree (2)?
D. (k=-2)
घात (2) के लिए \(x^3\) का गुणांक शून्य होना चाहिए और \(x^2\) का गुणांक अशून्य रहना चाहिए। (k+2=0) से (k=-2) मिलता है। / For degree (2), the coefficient of \(x^3\) must be zero and the coefficient of \(x^2\) must remain non-zero. From (k+2=0), (k=-2).
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If (p(x)=0x-6+5x-4-2x-3+9), what is the degree of (p(x))?
A. (4)
\(0x^6\) शून्य पद है और घात तय नहीं करता। सबसे बड़ी अशून्य घात (4) है। / \(0x^6\) is a zero term and does not determine the degree. The highest non-zero power is (4).
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Which expression is not a polynomial in (x), even though it has real coefficients?
C. \(7x^2+\frac{4}{x}-9\)
\(\frac{4}{x}=4x^{-1}\) है, जिसमें (x) की घात ऋणात्मक है। वास्तविक गुणांक होना अकेले बहुपद की गारंटी नहीं देता। / \(\frac{4}{x}=4x^{-1}\), where the power of (x) is negative. Having real coefficients alone does not guarantee a polynomial.
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If (p(x)=(a-4)x-5+3x-2-7) is to be exactly a quadratic polynomial, what is the value of (a)?
B. (4)
ठीक द्विघात के लिए \(x^5\) का गुणांक शून्य होना चाहिए। (a-4=0) से (a=4) मिलता है। / For exactly quadratic, the coefficient of \(x^5\) must be zero. From (a-4=0), (a=4).
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